Pith. sign in

REVIEW 3 major objections 5 minor 297 references

This paper claims that momentum after a stochastic Ehrenfest collapse should be rescaled in the branching plane of the effective coupling vector d_eff and gradient-difference vector g_eff, which reproduces benchmark populations and phase-sp

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-01 12:26 UTC pith:ZHPMET66

load-bearing objection New energy-gap-weighted d_eff is a real derivation and the branching-plane rescale is a practical improvement, but the benchmarks don't yet test the multi-state regime where the plane rotates — send it to review. the 3 major comments →

arxiv 2607.19536 v1 pith:ZHPMET66 submitted 2026-07-21 physics.chem-ph

Momentum Rescaling for Collapse to a Block Ehrenfest Dynamics

Ari R. X. Pereira , Benjamin G. Levine This is my paper
classification physics.chem-ph
keywords Ehrenfest dynamicscollapse to a blockmomentum rescalingbranching planeeffective nonadiabatic couplingdecoherence correctionconical intersectionnonadiabatic molecular dynamics
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper addresses a practical problem in the Ehrenfest-with-collapse-to-a-block (TAB) method: when the electronic density matrix stochastically collapses to a block, the nuclear momentum must be adjusted to conserve energy, and the direction of that adjustment matters. The authors show that rescaling along the instantaneous effective nonadiabatic coupling vector, d_eff, which is the formally correct impulse direction, fails in practice because collapse events happen well after population transfer and d_eff has rotated away from the direction that drove the transfer. They instead propose rescaling the component of momentum lying in the branching plane spanned by d_eff and the gradient-difference vector g_eff. In tests on a fulvene model and a three-state conical-intersection model, this branching-plane prescription reproduces benchmark populations and phase-space distributions while avoiding the anomalous mode excitation seen with other choices. The paper recommends this as the collapse-rescaling prescription for TAB and related stochastic Ehrenfest methods.

Core claim

The central claim is that the momentum rescaling direction at a TAB collapse should be the branching plane defined by d_eff and g_eff, not d_eff alone and not the momentum vector. The paper derives d_eff as the leading extra term in the effective force at a collapse, an energy-gap-weighted combination of nonadiabatic couplings between final and residual subspaces, and shows it can be computed on the fly without explicit couplings from force expectation values. Because TAB collapses occur stochastically after coherence has decayed, the instantaneous d_eff at the collapse can be rotated relative to the coupling direction at the earlier population-transfer time; the paper argues this rotation i

What carries the argument

The key object is the effective nonadiabatic coupling vector d_eff, defined as an energy-gap-weighted sum of coupling vectors connecting the final and residual subspaces, evaluated without explicit nonadiabatic couplings via force expectation values on the initial, final, and residual states. Together with the gradient-difference vector g_eff it defines the branching plane in which momentum is rescaled. This plane carries the argument because it lets the rescaling retain the direction of the coupling at the earlier population-transfer time, which can rotate away from the instantaneous d_eff by the collapse time.

Load-bearing premise

The prescription assumes that the coupling direction at the earlier population-transfer time is captured by the instantaneous branching plane spanned by d_eff and g_eff at the collapse time; the paper states this rotation is confined to the branching plane (exactly so in the two-state case) but does not prove it for multi-state systems with multiple coupling modes, and it deliberately drops first-order nonlocal terms in the force expansion.

What would settle it

Run the branching-plane rescaling on a model where the coupling direction at the time of population transfer lies largely outside the instantaneous branching plane at the collapse time, for example a three-state system with two distinct coupling modes and different state-pair gaps. If benchmark populations or phase-space distributions degrade, or if momentum distributions show spurious excitation, the claim that the relevant direction is captured by the instantaneous d_eff-g_eff plane is falsified.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Branching-plane rescaling should replace rescaling along the momentum direction or along d_eff in TAB simulations, giving reference-quality populations and phase-space distributions on the tested systems.
  • This approach preserves size consistency, which isotropic momentum rescaling does not.
  • d_eff can be computed on the fly without explicit nonadiabatic coupling vectors, requiring only one additional force evaluation on the residual state.
  • d_eff carries energy-gap weights, so it differs from previously proposed effective couplings whenever multiple state pairs with different gaps contribute; this matters beyond two-state systems.
  • The prescription is expected to transfer to other stochastic Ehrenfest methods, not just TAB.

Where Pith is reading between the lines

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

  • The rotation argument suggests that any trajectory method with delayed decoherence events could benefit from carrying a memory of the coupling direction at the population-transfer time; the branching plane is a minimal, memory-free proxy for that direction.
  • The most risky extension is to systems where the earlier population transfer is driven by a state pair whose coupling direction lies largely outside the instantaneous d_eff-g_eff plane at collapse; such cases may require pair-resolved rescaling.
  • The energy-gap weighting makes d_eff state-pair-specific, so an ensemble-level test comparing trajectories with identical histories but different collapse times could isolate the rotation effect and directly probe the branching-plane assumption.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes a momentum-rescaling prescription for the TAB (collapse-to-a-block Ehrenfest) method. After deriving an effective nonadiabatic coupling vector d_eff from a first-order Pechukas-force expansion for the instantaneous collapse of a superposition into a block subspace (Eqs. 13–31), the authors compare three rescaling directions: isotropic along p, along d_eff, and within the branching plane spanned by d_eff and g_eff (Eq. 42). Tested on the fulvene LVC model (vs. MCTDH) and a 3-state, 3-mode model with two conical intersections (vs. numerically exact grid TDSE), the branching-plane rule matches benchmark populations and phase-space distributions, whereas rescaling along d_eff produces spurious coupling-mode excitation. The paper also gives a force-evaluation-only expression for d_eff (Eq. 40) that avoids explicit nonadiabatic couplings.

Significance. If the branching-plane rule is valid beyond the fixed-plane LVC setting, it provides a practical, size-consistent collapse-rescaling direction for TAB and related stochastic Ehrenfest methods. The derivation of d_eff is careful and parameter-free; the benchmarks are independent exact/MCTDH references; and the code is released. The empirical improvement over d_eff rescaling is clear in the two models. However, the central geometric assumption—that the rotation of d_eff between population transfer and collapse is confined to the instantaneous branching plane—is asserted without proof and currently tested only in models where the coupling geometry is effectively fixed or collinear. The paper's contribution is therefore a promising heuristic plus a rigorous derivation of d_eff, not yet a fully established general prescription.

major comments (3)
  1. [§2.3.3, Eq. 42] The branching-plane rule is the central recommendation, but it is not a consequence of the Pechukas expansion. The statement that the rotation of d_eff 'is confined to the branching plane (exactly so in the two-state case)' is only exactly true when the g-h plane is constant (e.g., an LVC Hamiltonian). For a general two-state Hamiltonian the instantaneous branching plane rotates as R changes, and d_eff at the earlier population-transfer time can have a component perpendicular to the collapse-time plane; for more than two states there is no exact 2D branching plane at all. Since the first-order terms δd_eff, δd'_eff, δF_r, F_φ in Eq. 31 are dropped, the rule's domain of validity is unsupported. Please either prove or test the confinement assumption, or explicitly qualify the prescription as an approximation whose range needs mapping.
  2. [§3.2, Eq. 44] The 3-state benchmark does not exercise non-parallel coupling directions. In Eq. 44, states 1 and 2 are coupled to state 0 through the same mode x2 with identical constants (0.025), and H12=0, so all relevant d_ij vectors are parallel. Consequently the branching plane is essentially the same fixed x1-x2 plane throughout, and the test cannot detect a failure when different state pairs have non-collinear coupling vectors or when the plane rotates. The headline claim that the branching-plane rule 'reproduces benchmark populations on both test systems' is therefore established only for fixed-plane/collinear-coupling cases. Please add a benchmark with two distinct coupling modes (e.g., H01∝x2, H02∝x3) or a multi-state LVC model with nonparallel g-h planes, and report populations, momentum distributions, and frustrated-collapse statistics.
  3. [§2.2, Eq. 31] The decision to drop the O(Δt) terms in Eq. 31 is not quantitatively justified. Because the TAB collapse is instantaneous, Δt is a fictitious expansion parameter; the first-order corrections can be made arbitrarily small by taking Δt→0, so their absence cannot by itself justify neglecting nonlocal rotation of d_eff. If the relevant nonlocal timescale is the coherence-decay time τ_ij from Eq. 4, then the size of the dropped terms should be estimated in terms of τ_ij and the rate of change of d_eff along the trajectory. As written, the claim that 'the dominant neglected effect is the rotation of d_eff within the branching plane' is an assumption, not a derived consequence.
minor comments (5)
  1. [Eq. 39 and Eq. 42] When u=0 (e.g., p is perpendicular to the branching plane), Eq. 39 is undefined. The algorithm should specify a fallback convention, such as treating the collapse as frustrated, to avoid division by |u|.
  2. [§2.3.3] The term 'branching plane' is generalized to non-adiabatic final/residual states; this should be stated at first use rather than only in the Results section.
  3. [Figure 1] The blue d_eff arrows are hard to distinguish in monochrome print; consider color/arrow-style changes. The lower panel's 'Adiabatic Population' should define which adiabatic state is plotted.
  4. [Eq. 46] The notation d_old_eff is informal; a descriptive name would be clearer.
  5. [Text] Minor typographical issues: 'Y ork' in the affiliation and the undefined symbol pfrac in the Figure 5 caption; the caption should state pfrac = p·u/(|p||u|).

Circularity Check

0 steps flagged

No circularity: the deff derivation is a parameter-free perturbative expression and the branching-plane prescription is validated against external benchmarks; the unproved rotation assumption is a validity limitation, not a circular step.

full rationale

The central derivation chain is not circular. The effective nonadiabatic coupling vector deff is obtained from the Pechukas-force expansion (Eqs. 13–31) with no fitted constants; Eq. 40 is an algebraic identity relating deff to mean-field force expectation values via the decomposition psi_initial = sqrt(P) psi_final + sqrt(1-P) psi_residual, not an imposition of the target populations. The branching-plane rescaling (Eq. 42) is a fixed, parameter-free algorithm built from deff and geff; it is not tuned to reproduce the MCTDH or numerically exact TDSE benchmarks, which are external references (refs 51, 68). Self-citations to earlier TAB work (refs 23–26, 37) provide background methodology and the earlier p-rescaling choice, but the paper's new claim does not reduce to those citations or to a uniqueness theorem. The paper itself admits in §2.3.3 that the first-order terms of Eq. 31 are dropped and that the rotation of deff is assumed 'confined to the branching plane (exactly so in the two-state case)'; this is an explicit, honest statement of a heuristic whose multi-state validity is not proven. That is a correctness/domain-of-validity concern, not circularity, because the benchmarks remain independent and no input is re-described as a prediction. Therefore no circular step can be exhibited.

Axiom & Free-Parameter Ledger

1 free parameters · 5 axioms · 0 invented entities

The central claim rests on a first-order perturbative expansion of the Pechukas force (standard semiclassical approximation) plus the heuristic that the collapse-relevant coupling direction is preserved in the instantaneous branching plane. The only algorithmic free parameter carried in from prior TAB work is the decoherence-width α_N; nothing is fitted to the reference data in this paper. No new physical entities (particles, forces, dimensions) are introduced.

free parameters (1)
  • α_N (decoherence width parameter in Eq. 4) = Set to the position-space width of the initial wavepacket (e.g., σ_x = 0.204 a.u. in the 3-state model)
    Appears in the Bittner–Rossky decoherence time used by TAB. It is not fitted to the benchmarks in this paper but is an inherited user-set parameter from prior TAB work (refs 25, 37) that controls the collapse rate and thus the frequency of momentum rescaling events.
axioms (5)
  • standard math The Pechukas stationary-phase approximation gives the exact effective force on nuclei during a nonadiabatic transition (Eq. 9).
    Basis of the entire d_eff derivation; cited to Pechukas 1969 (refs 49, 50) and Coker–Xiao 1995 (ref 48). Assumed valid without independent re-derivation.
  • domain assumption First-order perturbation theory in Δt for the back-propagated electronic coefficients (Eq. 15) is sufficient, and leakage into states outside the initial subspace is negligible.
    Makes the expansion tractable; the paper states that neglecting leakage outside I is consistent with the Ehrenfest approximation.
  • domain assumption The collapse can be treated as an instantaneous resolution of |ψ_initial> into |ψ_final> and |ψ_residual> over a fictitious interval [t, t+Δt].
    Assumes locality in time of the collapse event, even though the underlying decoherence (Eq. 2) is continuous. The authors acknowledge this and retain only zeroth-order terms.
  • ad hoc to paper The dominant neglected effect after dropping first-order terms is the rotation of d_eff within the branching plane spanned by d_eff and g_eff.
    This is the key justification for the branching-plane rescaling (Eq. 42). It is exact in the two-state limit but assumed to hold for multi-state systems without proof; the paper itself says the rotation is 'confined to the branching plane (exactly so in the two-state case)'.
  • domain assumption The zeroth-order term -F_f is the Ehrenfest force already applied by the dynamics, so the collapse-specific impulse is along d_eff.
    Used in §2.3.2 to motivate rescaling along d_eff; the branching-plane variant then generalizes this direction to a two-dimensional subspace.

pith-pipeline@v1.3.0-alltime-deepseek · 13574 in / 10801 out tokens · 97535 ms · 2026-08-01T12:26:15.654748+00:00 · methodology

0 comments
read the original abstract

The Ehrenfest with collapse to a block (TAB) algorithm has recently been demonstrated to efficiently and accurately simulate nonadiabatic molecular dynamics in dense manifolds of electronic states. TAB employs an Ehrenfest force for the classical nuclei, accompanied by stochastic collapses of the electronic density matrix to account for decoherence. Energy conservation dictates that the nuclear momentum be adjusted during such a collapse. In this paper, we present a prescription for rescaling the component of the nuclear momentum that projects into the branching plane between arbitrary superposition states. This prescription yields accurate branching ratios and phase-space distributions when compared to alternative methods in which scaling is performed in the nonadiabatic coupling direction or the momentum direction. We justify this direction by deriving an effective nonadiabatic coupling vector from the localized Pechukas force during a collapse step.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

297 extracted references · 176 canonical work pages · 2 internal anchors

  1. [1]

    and Gross, E

    Abedi, Ali and Maitra, Neepa T. and Gross, E. K. U. , year = 2010, month = sep, journal =. Exact. doi:10.1103/PhysRevLett.105.123002 , urldate =

    work page doi:10.1103/physrevlett.105.123002 2010

  2. [2]

    Progress in Biophysics and Molecular Biology , volume =

    The Photochemical and Macromolecular Aspects of Vision , author =. Progress in Biophysics and Molecular Biology , volume =. doi:10.1016/0079-6107(67)90007-7 , urldate =

    work page doi:10.1016/0079-6107(67)90007-7

  3. [3]

    An Accurate, Non-Empirical Method for Incorporating Decoherence into

  4. [4]

    WIREs Computational Molecular Science , volume =

    Different Flavors of Nonadiabatic Molecular Dynamics , author =. WIREs Computational Molecular Science , volume =. doi:10.1002/wcms.1417 , urldate =

    doi:10.1002/wcms.1417

  5. [5]

    The Journal of Chemical Physics , volume =

    The Exact Forces on Classical Nuclei in Non-Adiabatic Charge Transfer , author =. The Journal of Chemical Physics , volume =. doi:10.1063/1.4908133 , urldate =

    work page doi:10.1063/1.4908133

  6. [6]

    Agostini, Federica , urldate =

  7. [7]

    The European Physical Journal B , volume =

    Nuclear Quantum Effects in Electronic (Non)Adiabatic Dynamics , author =. The European Physical Journal B , volume =. doi:10.1140/epjb/e2018-90144-3 , urldate =

    doi:10.1140/epjb/e2018-90144-3

  8. [8]

    Agostini, Federica and Curchod, Basile F. E. , year = 2026, month = apr, journal =. A Pictorial (and Hopefully Pedagogical) Discussion on the. doi:10.1007/s10698-025-09559-9 , urldate =

    work page doi:10.1007/s10698-025-09559-9 2026

  9. [9]

    Agostini, Federica and Min, Seung Kyu and Abedi, Ali and Gross, E. K. U. , year = 2016, journal =. Quantum-Classical Nonadiabatic Dynamics: Coupled-vs Independent-Trajectory Methods , shorttitle =

    2016

  10. [10]

    and Long, Run and Prezhdo, Oleg V

    Akimov, Alexey V. and Long, Run and Prezhdo, Oleg V. , year = 2014, month = may, journal =. Coherence Penalty Functional:. doi:10.1063/1.4875702 , urldate =

    work page doi:10.1063/1.4875702 2014

  11. [11]

    and Warren, Patrick B

    Allen, Rosalind J. and Warren, Patrick B. and Ten Wolde, Pieter Rein , year = 2005, month = jan, journal =. Sampling. doi:10.1103/PhysRevLett.94.018104 , urldate =

    doi:10.1103/physrevlett.94.018104 2005

  12. [12]

    Alonso, J. L. and. Ehrenfest Dynamics Is Purity Non-Preserving:. The Journal of Chemical Physics , volume =. doi:10.1063/1.4737861 , urldate =

    work page doi:10.1063/1.4737861

  13. [13]

    Alonso, J. L. and Castro, A. and. Statistics and. Journal of Physics A: Mathematical and Theoretical , volume =. doi:10.1088/1751-8113/44/39/395004 , urldate =

    work page doi:10.1088/1751-8113/44/39/395004

  14. [14]

    Molecular

    Amdursky, Nadav and Erez, Yuval and Huppert, Dan , year = 2012, month = sep, journal =. Molecular. doi:10.1021/ar300053p , urldate =

    work page doi:10.1021/ar300053p 2012

  15. [15]

    Statistical

    Amit, Daniel J and Verbin, Yosef , year = 1999, month = dec, publisher =. Statistical. doi:10.1142/3526 , urldate =

    work page doi:10.1142/3526 1999

  16. [16]

    Ardani, Kristen and Denholm, Paul and Mai, Trieu and Margolis, Robert and O'Shaughnessy, Eric and Silverman, Timothy and Zuboy, Jarett , year = 2021, month = aug, publisher =. Solar

    2021

  17. [17]

    Electronic Coherences in Molecules: The Projected Nuclear Quantum Momentum as a Hidden Agent

    Arribas, Evaristo Villaseco and Maitra, Neepa T. , year = 2024, month = jun, number =. Electronic. doi:10.48550/arXiv.2405.00649 , urldate =. arXiv , keywords =:2405.00649 , publisher =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2405.00649 2024

  18. [18]

    , year = 2024, month = dec, journal =

    Arribas, Evaristo Villaseco and Maitra, Neepa T. , year = 2024, month = dec, journal =. Electronic. doi:10.1103/PhysRevLett.133.233201 , urldate =

    work page doi:10.1103/physrevlett.133.233201 2024

  19. [19]

    Journal of Chemical Theory and Computation , volume =

    Nudged. Journal of Chemical Theory and Computation , volume =. doi:10.1021/acs.jctc.1c00462 , urldate =

    doi:10.1021/acs.jctc.1c00462

  20. [20]

    Bachmann, Paul Gustav Heinrich , year = 1894, publisher =

  21. [21]

    doi:10.1142/9789812839664_bmatter , urldate =

    Classical and. doi:10.1142/9789812839664_bmatter , urldate =

    work page doi:10.1142/9789812839664_bmatter

  22. [22]

    and Lesyng, B

    Bala, P. and Lesyng, B. and McCammon, J. A. , year = 1994, month = mar, journal =. Extended. doi:10.1016/0009-2614(94)87055-1 , urldate =

    work page doi:10.1016/0009-2614(94)87055-1 1994

  23. [23]

    Nature Energy , volume =

    Defects in Perovskite-Halides and Their Effects in Solar Cells , author =. Nature Energy , volume =. doi:10.1038/nenergy.2016.149 , urldate =

    doi:10.1038/nenergy.2016.149 2016

  24. [24]

    Bank, RCSB Protein Data , journal =

  25. [25]

    WIREs Computational Molecular Science , volume =

    Nonadiabatic Dynamics with Trajectory Surface Hopping Method , author =. WIREs Computational Molecular Science , volume =. doi:10.1002/wcms.64 , urldate =

    doi:10.1002/wcms.64

  26. [26]

    Relaxation Mechanisms of

    Barbatti, Mario and Aquino, Ad. Relaxation Mechanisms of. Proceedings of the National Academy of Sciences , volume =. doi:10.1073/pnas.1014982107 , urldate =

    work page doi:10.1073/pnas.1014982107

  27. [27]

    Velocity

    Barbatti, Mario , year = 2021, month = may, journal =. Velocity. doi:10.1021/acs.jctc.1c00012 , urldate =

    doi:10.1021/acs.jctc.1c00012 2021

  28. [28]

    Physical Review A , volume =

    Nonequilibrium Effects on One-Norm Geometric Correlations and the Emergence of a Pointer-State Basis in the Weak- and Strong-Coupling Regimes , author =. Physical Review A , volume =. doi:10.1103/PhysRevA.104.042417 , urldate =

    work page doi:10.1103/physreva.104.042417

  29. [29]

    Multiconfigurational Quantum Chemistry:

    Battaglia, Stefano and Galv. Multiconfigurational Quantum Chemistry:. doi:10.1016/B978-0-323-91738-4.00016-6 , urldate =. arXiv , langid =:2305.06678 , primaryclass =

    Pith/arXiv arXiv

  30. [30]

    Nature Communications , volume =

    E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials , author =. Nature Communications , volume =. doi:10.1038/s41467-022-29939-5 , urldate =

    doi:10.1038/s41467-022-29939-5

  31. [31]

    The Journal of Chemical Physics , volume =

    Mean-Field Dynamics with Stochastic Decoherence (. The Journal of Chemical Physics , volume =. doi:10.1063/1.2131056 , urldate =

    work page doi:10.1063/1.2131056

  32. [32]

    Projections of Quantum Observables onto Classical Degrees of Freedom in Mixed Quantum/Classical Simulations:

  33. [34]

    The Journal of Physical Chemistry A , volume =

    Ab. The Journal of Physical Chemistry A , volume =. doi:10.1021/jp994174i , urldate =

    doi:10.1021/jp994174i

  34. [35]

    and Reichman, David R

    Berkelbach, Timothy C. and Reichman, David R. and Markland, Thomas E. , year = 2012, month = jan, journal =. Reduced Density Matrix Hybrid Approach:. doi:10.1063/1.3671372 , urldate =

    work page doi:10.1063/1.3671372 2012

  35. [36]

    and Markland, Thomas E

    Berkelbach, Timothy C. and Markland, Thomas E. and Reichman, David R. , year = 2012, month = feb, journal =. Reduced Density Matrix Hybrid Approach:. doi:10.1063/1.3687342 , urldate =

    work page doi:10.1063/1.3687342 2012

  36. [37]

    Classical and

    Berne, Bruce J and Ciccotti, Giovanni and Coker, David F , year = 1998, month = jun, publisher =. Classical and. Classical and

    1998

  37. [38]

    , year = 1998, month = jun, pages =

    Berne, Bruce J. , year = 1998, month = jun, pages =. Molecular Dynamics Methods for the Enhanced Sampling of Phase Space , booktitle =. doi:10.1142/9789812839664_0006 , urldate =

    work page doi:10.1142/9789812839664_0006 1998

  38. [39]

    Berne, Bruce J and Ciccotti, Giovanni and Coker, David F , year = 1998, month = jun, publisher =. [. Classical and

    1998

  39. [40]

    Bernstein, Jeremy , year = 2018, month = apr, publisher =. A. doi:10.1142/10762 , urldate =

    work page doi:10.1142/10762 2018

  40. [41]

    Berry, Michael , year = 1988, journal =. The. 24989302 , eprinttype =

    1988

  41. [42]

    Molecular Mechanism of

    Biancalana, Matthew and Koide, Shohei , year = 2010, month = jul, journal =. Molecular Mechanism of. doi:10.1016/j.bbapap.2010.04.001 , urldate =

    work page doi:10.1016/j.bbapap.2010.04.001 2010

  42. [43]

    Chemical Physics Reviews , volume =

    The Phase-Space Way to Electronic Structure Theory and Subsequently Chemical Dynamics , author =. Chemical Physics Reviews , volume =. doi:10.1063/5.0286240 , urldate =

    doi:10.1063/5.0286240

  43. [45]

    Biochimica et Biophysica Acta (BBA) - Bioenergetics , volume =

    Nature of the Primary Photochemical Events in Rhodopsin and Bacteriorhodopsin , author =. Biochimica et Biophysica Acta (BBA) - Bioenergetics , volume =. doi:10.1016/0005-2728(90)90163-X , urldate =

    work page doi:10.1016/0005-2728(90)90163-x

  44. [46]

    The Journal of Chemical Physics , volume =

    Decoherent Histories and Nonadiabatic Quantum Molecular Dynamics Simulations , author =. The Journal of Chemical Physics , volume =. doi:10.1063/1.475013 , urldate =

    work page doi:10.1063/1.475013

  45. [47]

    and Rossky, Peter J

    Bittner, Eric R. and Rossky, Peter J. , year = 1995, month = nov, journal =. Quantum Decoherence in Mixed Quantum-classical Systems:. doi:10.1063/1.470177 , urldate =

    work page doi:10.1063/1.470177 1995

  46. [48]

    , year = 2010, publisher =

    Bittner, Eric R. , year = 2010, publisher =. Quantum Dynamics: Applications in Biological and Materials Systems , shorttitle =

    2010

  47. [49]

    , year = 2010, publisher =

    Bittner, Eric R. , year = 2010, publisher =. Quantum. Quantum

    2010

  48. [50]

    Blechman, Andrew E , year = 2023, month = may, publisher =. A. doi:10.1142/10933 , urldate =

    work page doi:10.1142/10933 2023

  49. [51]

    Europhysics Letters (EPL) , volume =

    Generalized Coherent States as Preferred States of Open Quantum Systems , author =. Europhysics Letters (EPL) , volume =. doi:10.1209/0295-5075/79/40003 , urldate =

    work page doi:10.1209/0295-5075/79/40003

  50. [52]

    doi:10.1002/jcc.20494 , urldate =

    Bond Order and Valence Indices:. doi:10.1002/jcc.20494 , urldate =

    work page doi:10.1002/jcc.20494

  51. [53]

    Dynamical

    Born, Max and Huang, Kun , year = 1998, month = nov, series =. Dynamical

    1998

  52. [54]

    The Journal of Chemical Physics , volume =

    Quantum-Classical Molecular Dynamics as an Approximation to Full Quantum Dynamics , author =. The Journal of Chemical Physics , volume =. doi:10.1063/1.471952 , urldate =

    work page doi:10.1063/1.471952

  53. [55]

    and Oppenheimer, R

    Born, M. and Oppenheimer, R. , year = 1927, journal =. Zur. doi:10.1002/andp.19273892002 , urldate =

    doi:10.1002/andp.19273892002 1927

  54. [56]

    and Delacr

    Broyer, M. and Delacr. Stimulated Emission Spectroscopy of the Ground State of. The Journal of Chemical Physics , volume =. doi:10.1063/1.456599 , urldate =

    work page doi:10.1063/1.456599

  55. [57]

    Science , volume =

    Ultrafast Electron Dynamics in Phenylalanine Initiated by Attosecond Pulses , author =. Science , volume =. doi:10.1126/science.1254061 , urldate =

    doi:10.1126/science.1254061

  56. [58]

    Carenza, Livio Nicola and. Quasi-. Physical Review Letters , volume =. doi:10.1103/PhysRevLett.134.128304 , urldate =

    doi:10.1103/physrevlett.134.128304

  57. [59]

    Barrier Crossings: Classical Theory of Rare but Important Events , shorttitle =

    Chandler, David , year = 1998, month = jun, pages =. Barrier Crossings: Classical Theory of Rare but Important Events , shorttitle =. Classical and. doi:10.1142/9789812839664_0001 , urldate =

    work page doi:10.1142/9789812839664_0001 1998

  58. [60]

    Electron Transfer in Water and Other Polar Environments, How It Happens , booktitle =

    Chandler, David , year = 1998, month = jun, pages =. Electron Transfer in Water and Other Polar Environments, How It Happens , booktitle =. doi:10.1142/9789812839664_0002 , urldate =

    work page doi:10.1142/9789812839664_0002 1998

  59. [61]

    Finding Transition Pathways: Throwing Ropes over Rough Mountain Passes, in the Dark , shorttitle =

    Chandler, David , year = 1998, month = jun, pages =. Finding Transition Pathways: Throwing Ropes over Rough Mountain Passes, in the Dark , shorttitle =. Classical and. doi:10.1142/9789812839664_0003 , urldate =

    work page doi:10.1142/9789812839664_0003 1998

  60. [62]

    Methods to

    Chen, Hsing-Ta and Chen, Junhan and. Methods to. Journal of Chemical Theory and Computation , volume =. doi:10.1021/acs.jctc.1c01304 , urldate =

    doi:10.1021/acs.jctc.1c01304

  61. [63]

    and Yang, Chen and Zheng, Xiaopeng and Gereige, Issam and AlSaggaf, Ahmed and Mohammed, Omar F

    Chen, Zhaolai and Turedi, Bekir and Alsalloum, Abdullah Y. and Yang, Chen and Zheng, Xiaopeng and Gereige, Issam and AlSaggaf, Ahmed and Mohammed, Omar F. and Bakr, Osman M. , year = 2019, month = jun, journal =. Single-. doi:10.1021/acsenergylett.9b00847 , urldate =

    work page doi:10.1021/acsenergylett.9b00847 2019

  62. [64]

    , year = 2021, month = mar, journal =

    Christopoulou, Georgia and Freibert, Antonia and Worth, Graham A. , year = 2021, month = mar, journal =. Improved Algorithm for the Direct Dynamics Variational Multi-Configurational. doi:10.1063/5.0043720 , urldate =

    work page doi:10.1063/5.0043720 2021

  63. [65]

    Achieving the

    Christy, I-Juliana , year = 2013, month = jul, publisher =. Achieving the. doi:10.1142/8756 , urldate =

    work page doi:10.1142/8756 2013

  64. [66]

    Constrained and Nonequilibrium Molecular Dynamics , booktitle =

    Ciccotti, Giovanni and Ferrario, Mauro , year = 1998, month = jun, pages =. Constrained and Nonequilibrium Molecular Dynamics , booktitle =. doi:10.1142/9789812839664_0007 , urldate =

    work page doi:10.1142/9789812839664_0007 1998

  65. [67]

    The Journal of Chemical Physics , volume =

    Methods for Molecular Dynamics with Nonadiabatic Transitions , author =. The Journal of Chemical Physics , volume =. doi:10.1063/1.469428 , urldate =

    work page doi:10.1063/1.469428

  66. [68]

    and Mei, Hsiao S

    Coker, David F. and Mei, Hsiao S. and Ryckaert, Jean-Paul , year = 1998, month = jun, pages =. Thermal Average Time Correlation Functions from Non-Adiabatic. Classical and. doi:10.1142/9789812839664_0023 , urldate =

    work page doi:10.1142/9789812839664_0023 1998

  67. [69]

    Reviews of Modern Physics , volume =

    Entropic Uncertainty Relations and Their Applications , author =. Reviews of Modern Physics , volume =. doi:10.1103/RevModPhys.89.015002 , urldate =

    doi:10.1103/revmodphys.89.015002

  68. [70]

    The Complete Active Space

  69. [71]

    Cornelio, M. F. and Far. Emergence of the. Physical Review Letters , volume =. doi:10.1103/PhysRevLett.109.190402 , urldate =

    work page doi:10.1103/physrevlett.109.190402

  70. [72]

    Chemical Reviews , volume =

    Recent. Chemical Reviews , volume =. doi:10.1021/acs.chemrev.7b00577 , urldate =

    doi:10.1021/acs.chemrev.7b00577

  71. [73]

    Nature Energy , volume =

    The Underestimated Potential of Solar Energy to Mitigate Climate Change , author =. Nature Energy , volume =. doi:10.1038/nenergy.2017.140 , urldate =

    work page doi:10.1038/nenergy.2017.140 2017

  72. [74]

    Intrinsic and Light-Induced Nonadiabatic Phenomena in the

    Csehi, Andr. Intrinsic and Light-Induced Nonadiabatic Phenomena in the. Physical Chemistry Chemical Physics , volume =. doi:10.1039/C7CP02164J , urldate =

    doi:10.1039/c7cp02164j

  73. [75]

    arXiv , langid =:1104.4405 , primaryclass =

    Time-Dependent Pointer States and Determination of the Preferred Basis of Measurement , author =. arXiv , langid =:1104.4405 , primaryclass =

    Pith/arXiv arXiv

  74. [76]

    De Carvalho, Felipe Franco and Bouduban, Marine E. F. and Curchod, Basile F. E. and Tavernelli, Ivano , year = 2014, month = jan, journal =. Nonadiabatic. doi:10.3390/e16010062 , urldate =

    work page doi:10.3390/e16010062 2014

  75. [77]

    Journal of Physics A: Mathematical and General , volume =

    Minimal Uncertainty States for the Rotation and Allied Groups , author =. Journal of Physics A: Mathematical and General , volume =. doi:10.1088/0305-4470/10/11/012 , urldate =

    work page doi:10.1088/0305-4470/10/11/012

  76. [78]

    and Thorson, Walter R

    Delos, John B. and Thorson, Walter R. and Knudson, Stephen K. , year = 1972, month = aug, journal =. Semiclassical. doi:10.1103/PhysRevA.6.709 , urldate =

    work page doi:10.1103/physreva.6.709 1972

  77. [79]

    Science , volume =

    Impact of Microstructure on Local Carrier Lifetime in Perovskite Solar Cells , author =. Science , volume =. doi:10.1126/science.aaa5333 , urldate =

    work page doi:10.1126/science.aaa5333

  78. [80]

    arXiv , langid =:2110.03235 , primaryclass =

    Joint. arXiv , langid =:2110.03235 , primaryclass =

    Pith/arXiv arXiv

  79. [81]

    and Yarkony, David R

    Diffenderfer, Randall N. and Yarkony, David R. , year = 1982, month = dec, journal =. Use of the State-Averaged. doi:10.1021/j100223a010 , urldate =

    work page doi:10.1021/j100223a010 1982

  80. [82]

    Decoherent

    Di. Decoherent. Physical Review Letters , volume =. doi:10.1103/PhysRevLett.74.203 , urldate =

    work page doi:10.1103/physrevlett.74.203

Showing first 80 references.