A coupled-trajectory approach for decoherence, frustrated hops and internal consistency in surface hopping

Eduarda Sangiogo Gil; Federica Agostini; Lea M. Ibele; Peter Sch\"urger; Rodrigue Noc

REVIEW 3 major objections 5 minor 13 references

A coupled-trajectory approach for decoherence, frustrated hops and internal consistency in surface hopping

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Treating the trajectory swarm as a single energy-sharing entity yields a surface hopping scheme that restores internal consistency, removes frustrated hops, and matches quantum reference calculations in two molecular tests.

desk verdict Energy-sharing surface hopping that fixes frustrated hops and internal consistency on two LVC benchmarks, but the coupled-trajectory mechanism is oversold. read the letter →

arxiv 2412.04958 v3 pith:4MMZN4S6 submitted 2024-12-06 physics.chem-ph

Lea M. Ibele , Eduarda Sangiogo Gil , Peter Schürger , Rodrigue Noc , Federica Agostini This is my paper · ORCID

classification physics.chem-ph

keywords surfacehoppingnonadiabaticmoleculardynamicscoupledtrajectoriesquantummomentumfrustratedhopsinternalconsistencydecoherenceexactfactorization

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Surface hopping is the workhorse of photochemical dynamics, but in its standard independent-trajectory form it overestimates electronic coherence, produces frustrated hops when a trajectory lacks kinetic energy to climb to the final state, and gives two inconsistent estimates of electronic populations. This paper claims the cure is to treat the swarm of trajectories as a single coupled object, not as independent copies. The proposed coupled-trajectory surface hopping scheme adds a quantum-momentum term that carries nuclear delocalization into the electronic evolution, and when a hop is attempted it lets the whole swarm share kinetic energy through three alternative rules (equity, overlap, quantum-momentum-weighted). On full-dimensional linear-vibronic-coupling models of fulvene and DMABN, the equity and overlap versions remove frustrated hops, restore internal consistency, and give diabatic populations in good agreement with converged quantum reference calculations.

What carries the argument

The central object is the coupled-trajectory surface hopping scheme with energy sharing, built from the exact factorization of the molecular wavefunction. The nuclear swarm is used to reconstruct the nuclear density, and the gradient of that density enters the electronic coefficient evolution as a quantum momentum term, which is the decoherence channel. The load-bearing mechanism for the other two issues is energy sharing: at a hop attempt, the hopping trajectory first gives all of its kinetic energy, and the remaining deficit is redistributed to the other trajectories by rescaling their velocities either proportionally to their kinetic energy (equity) or proportional to their Gaussian overlap with the hopping trajectory (overlap), with a third option splitting the cost by nonadiabatic-coupling and quantum-momentum weights. The active state can also be chosen deterministically as the state of largest electronic population rather than by a stochastic fewest-switches rule.

What would settle it

Run the overlap-based energy-sharing scheme on a model in which the nuclear density separates into two well-separated branches that later re-enter a nonadiabatic coupling region; if the swarm populations deviate from a converged wavepacket calculation, or if trajectories in one branch share energy for hops occurring in the other, the local character of the hop is lost and the frozen-Gaussian quantum momentum is the component to blame.

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Extended reading notes

Core claim

The central claim is that moving away from an independent-trajectory picture is the one strategy that addresses decoherence, frustrated hops, and internal consistency simultaneously. In the new algorithm, each nuclear trajectory still evolves on an adiabatic potential and hops between electronic states, but the electronic coefficients are corrected by a coupled-trajectory term depending on the quantum momentum, and the active state can be chosen deterministically as the most populated state. At a hop attempt, if the cost exceeds the hopping trajectory's kinetic energy, the deficit is taken from the kinetic energy of other trajectories, so the swarm as a whole conserves energy and the hop is not frustrated. In the two molecular models tested, the equity- and overlap-based sharing schemes eliminate frustrated hops, make the trajectory-count and electronic-population estimates agree, and reproduce the reference diabatic population decay.

Load-bearing premise

The scheme stands or falls on the assumption that a sum of frozen Gaussians centered on the trajectories gives a reliable quantum momentum for the electronic evolution and the energy sharing, even though the paper notes this reconstruction violates a physical no-transfer condition and must be imposed by hand.

Editorial extensions

If this is right

In the fulvene model, the equity- and overlap-based schemes produce zero frustrated hops even when the velocity is rescaled only along the nonadiabatic coupling vector, where standard surface hopping records hundreds of frustrated hops.
CCT-TSH achieves internal consistency without an external decoherence correction: the fraction of trajectories on a state and the averaged electronic population agree.
On DMABN, the equity- and overlap-based schemes again remove frustrated hops and give S2 populations close to the decoherence-corrected surface hopping reference.
The overlap-based scheme with deterministic active-state selection matches the quantum reference diabatic populations in both molecules, indicating that deterministic hopping is a viable alternative to stochastic fewest-switches hopping.
Changing the velocity-rescaling scheme, or initializing the swarm with positions-only instead of Wigner sampling, changes the populations somewhat but does not destroy internal consistency or reintroduce frustrated hops.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The near-identical performance of the equity and overlap schemes suggests the essential ingredient may be swarm-level energy conservation rather than the specific sharing rule; a minimal scheme that pools kinetic energy from all trajectories could be tested directly.
The paper does not test cases where the swarm splits into well-separated spatial branches, so the local character of overlap-based sharing remains open; a model with two distant wavepacket branches crossing the same coupling region would stress it.
Because the quantum-momentum-based sharing still suffers many frustrated hops, the frozen-Gaussian reconstruction of the nuclear density is the most likely weak point; replacing it with an adaptive or higher-order density estimate is a natural next step.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things to know about this paper. It introduces something new: three schemes for sharing kinetic energy among a swarm of trajectories when a hop is attempted, plus a deterministic hopping rule based on the largest electronic population. And the equity- and overlap-based variants do deliver on the paper's main promises—no frustrated hops, internal consistency, and diabatic populations close to MCTDH on the fulvene and DMABN LVC benchmarks. But the paper advertises the coupled-trajectory quantum-momentum term as the key to overcoming all three problems at once, and that attribution is not actually supported by the tests. The energy reservoir and the deterministic hop rule could be doing most of the work; there is no control run that switches off Eq. (10) while keeping the rest.

What's genuinely good: the energy-sharing idea is a real departure from the usual per-trajectory velocity rescaling, and the paper is transparent about its heuristic character. The deterministic hop rule is clean and physically motivated from the exact-factorization picture. The implementation is in the open-source G-CTMQC code, the input generator is provided, and the benchmarks are external MCTDH results. The paper also openly reports that the quantum-momentum-based sharing scheme produces 120 frustrated hops and that the frozen-Gaussian quantum momentum violates a sum rule and needs an ad hoc patch.

The soft spots are real but not fatal. First, the missing ablation: run CCT-TSH with the same energy sharing and deterministic hops but without the C_CT term to see how much the quantum momentum actually contributes. If the results barely change, the abstract-level claim oversells the coupled-trajectory mechanism. Second, the frustrated-hop removal for equity/overlap is by construction, as the authors admit; it's a reservoir effect, not evidence that the nonlocal coupling works. Third, internal consistency is likely helped by the largest-population rule, which directly ties the active state to the propagated electronic populations. Fourth, there is no uncertainty analysis for CCT-TSH over the velocity-rescaling variants—they say results don't change dramatically but show no spread. Finally, both tests are LVC models; an ab initio or non-LVC test would make the robustness claim credible.

Who gets value from this: surface hopping practitioners who want a concrete way to eliminate frustrated hops and restore internal consistency without ad hoc decoherence corrections. A serious referee should be assigned; the paper is coherent, reproducible, and the method will likely be used regardless of whether the mechanism attribution survives scrutiny. My own verdict is conditional: publish after an ablation and a non-LVC test.

Referee Report

3 major / 5 minor

Summary. This manuscript proposes CCT-TSH, a surface hopping algorithm built from exact-factorization coupled-trajectory ideas. Electronic coefficients evolve via the standard Tully term plus a quantum-momentum coupling (Eq. 10); nuclei evolve on the active adiabatic surface; hops are chosen deterministically as the state with largest electronic population (or by fewest switches in a variant); and three schemes (equity, overlap, quantum-momentum) share kinetic energy among the swarm when a hop is energetically unfavorable. The method is tested on full-dimensional LVC models of fulvene (30 modes) and DMABN (57 modes). The equity and overlap schemes eliminate frustrated hops, restore near-agreement between electronic populations P and trajectory fractions F, and yield diabatic populations in good agreement with MCTDH reference data; the quantum-momentum scheme yields 120 frustrated hops in both systems and is less accurate.

Significance. If substantiated, the contribution is significant: it offers a practically implementable surface hopping variant in which internal consistency and absence of frustrated hops are achieved without a per-trajectory decoherence correction, and it connects the design to the exact factorization. The paper has concrete strengths: it uses an open-source implementation (G-CTMQC), provides a Supporting-Information script for generating inputs, uses external MCTDH benchmarks, and introduces no fitted parameters in the derivation. However, the evidence as presented does not cleanly attribute the improvements to the coupled-trajectory electronic term (Eq. 10); the deterministic hopping rule and collective energy reservoir can produce the same headline behaviors by construction, while the one scheme most directly tied to the quantum momentum performs worst. The manuscript therefore needs a control/ablation study and quantitative diagnostics before the broad claims in the abstract and conclusions can be accepted.

major comments (3)

[Section 3.2 / Table 1 / Fig. 2] The manuscript does not report a control calculation that isolates the role of the coupled-trajectory electronic term, Eq. (10), from the two algorithmic features that can enforce the advertised behavior by construction: the deterministic hopping rule ('trajectories evolve in the electronic state with the largest population', Section 2) and the collective energy-sharing reservoir. Equity- and overlap-based CCT-TSH eliminate frustrated hops 'by construction' (Section 3.2), and deterministic hopping directly aligns the trajectory fraction F with the electronic population P. Since the conclusions claim that coupling of trajectories is 'the crucial ingredient' (Section 4), the authors should add an ablation, e.g., CCT-TSH with Eq. (10) switched off while retaining deterministic hopping and energy sharing, or an independent-trajectory energy-sharing benchmark, to rule out the alternative explanation that the energy reservoir alone is responsible for the improvements.
[Section 3.2 / Table 1 / Fig. 6] The quantum-momentum-based energy-sharing scheme, which is the variant most directly built on the quantum momentum appearing in Eq. (10), produces 120 frustrated hops in both fulvene and DMABN and deviates from the other schemes in Fig. 6. The authors attribute this to the approximate nature of the quantum momentum (Sections 2 and 3.2). This is a load-bearing caveat: it shows that the coupled-trajectory electronic dynamics as implemented is not by itself the robust cure claimed in the abstract, and it weakens the attribution of the success of the equity/overlap schemes to the quantum-momentum coupling. Please discuss quantitatively the role of Eq. (10) in the successful schemes, for example by reporting the relative magnitude of the CT contribution to the electronic coefficient derivative and by comparing with the suggested ablation.
[Section 3.2 / Figs. 2, 5, 7] The paper reports no quantitative measure of internal consistency (e.g., the maximum or time-integrated absolute difference between F and P) and no diagnostic of total-energy conservation of the swarm, although energy conservation of the swarm is the basis of the energy-sharing schemes. Adding such diagnostics is necessary to support the qualitative statements that internal consistency is 'restored closely' and that the schemes give 'almost identical' population decays. Please include the corresponding error norms or energy-conservation curves.

minor comments (5)

[Abstract and Fig. 6 caption] There are typos: '4-(dimethyloamino)benzonitrile' in the Abstract should be '4-(dimethylamino)benzonitrile'; 'dfferent' in the Fig. 6 caption and 'occurences' in the Conclusions should be corrected.
[References 29 and 93] References 29 and 93 appear to cite the same paper by Gómez, Spinlove, and Worth with different publication years; please verify and cite it once to avoid confusion.
[Fig. 2] The caption states that the lighter lines show different velocity rescaling/inversion choices for TSH, TSH-ED, and CT-TSH, but it is not stated whether the CCT-TSH curves shown in the same figure are individual runs or averages; please clarify the color/line coding and report the spread over the three energy-sharing schemes.
[Eqs. (14) and (15)] The symbol q is introduced as a pseudo-velocity but is then used in Eqs. (14) and (15) without an explicit definition; please define it (e.g., q_alpha = Q_alpha/M) just before the equations and use the same notation consistently.
[Section 2 and Conclusions] The statement that the frozen-Gaussian reconstruction of the quantum momentum 'violates the physical conditions' and must be imposed a posteriori is an important limitation; it should be restated in the Conclusions so that readers who do not read Section 2 in detail are aware of this caveat.

Circularity Check

2 steps flagged · score 6.0 of 10

CCT-TSH's headline cures are partly built into the algorithm: deterministic largest-population hopping defines F≈P, and equity/overlap energy sharing defines away frustrated hops.

self definitional [Section 2, deterministic hopping paragraph; Section 3.2, Fig. 5 discussion]
"Internal consistency refers to the agreement between the two ways of estimating the population of the electronic states in surface hopping: the fraction of trajectories associated to, or evolving in, an electronic state (F); the trajectory-averaged population propagated along each trajectory (P)... in CCT-TSH, the trajectories evolve in the electronic state with the largest population."

Internal consistency is defined as agreement between the trajectory fraction F and the propagated population P. The CCT-TSH hopping rule selects as active state the electronic state with the largest |C|^2, so the trajectory label is set directly from the population dynamics. The resulting F≈P is therefore an input of the algorithm's definition rather than an emergent consequence of the coupled-trajectory term in Eq. (10). The paper presents this as the scheme restoring internal consistency, but the restoration is built into the deterministic hop rule; no test removes that rule to show that the CT coupling alone would produce the same F-P agreement.
self definitional [Section 3.2, numerical results for fulvene, Table 1 discussion]
"This issue is completely alleviated, albeit by construction, when using the equity-based and overlap-based CCT-TSH schemes."

In the equity and overlap schemes, whenever the hopping trajectory lacks kinetic energy, Eqs. (12)-(13) rescale other trajectories' velocities to supply Delta E_CCT_hop = Delta E_hop - E_kin. Thus a hop is frustrated only if the entire swarm cannot cover the deficit; zero frustrated hops in Table 1 is guaranteed by this bookkeeping, exactly as the paper concedes with 'albeit by construction.' The quantum-momentum-based scheme, which does not guarantee full collective compensation, records 120 frustrated hops, confirming that the zeros are a property of the sharing construction rather than of the coupled-trajectory electronic evolution.

full rationale

Two advertised improvements reduce to algorithmic construction rather than to the coupled-trajectory electronic dynamics. First, internal consistency is defined as F≈P, and CCT-TSH chooses the active state as the state with the largest electronic population, so the trajectory fraction is designed to follow the quantum population; this is a definitional alignment, not an emergent result. Second, the equity- and overlap-based energy-sharing schemes are defined to draw the hop deficit from the swarm, so zero frustrated hops in Table 1 is guaranteed by construction, as the paper explicitly concedes. The paper is nevertheless not globally circular: the diabatic populations are benchmarked against external MCTDH results, no parameters are fitted to those references, the Qmom variant still yields 120 frustrated hops and serves as a contrast, and the self-citations to CT-MQC/CT-TSH are methodological background rather than a uniqueness argument. The partial circularity is confined to the two headline properties of internal consistency and frustration-free hops; the quantitative population agreement retains independent content.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The method relies on several approximations inherited from the exact factorization literature: the frozen-Gaussian reconstruction of the nuclear density for the quantum momentum, purely adiabatic forces, and the new heuristic energy-sharing rules. Free parameters include the kinetic energy threshold for contributors, the population threshold 0.5, and the unspecified Gaussian width for overlap. No new physical entities are postulated.

free parameters (4)

kinetic energy threshold = not reported
Trajectories below this kinetic energy are excluded from contributing to energy sharing in equity and overlap schemes; value not specified in the text.
population threshold for deterministic hopping = 0.5
Trajectories switch to the electronic state with largest population, with the threshold chosen as 0.5 in the calculations.
overlap Gaussian width = not specified
Width used to compute overlap S^alpha beta between trajectories; the paper states the result is insensitive as long as widths are equal and large enough, but no value is reported.
TSH-ED energy parameter = 0.1 Ha
Standard value used for the energy-based decoherence correction in the TSH-ED baseline.

assumptions (5)

standard math Exact factorization of the molecular wavefunction (Eq. 1) is valid and the coupled equations (3)-(4) are exact.
Standard result from Abedi, Maitra, Gross (Refs 54-55), used as starting point.
domain assumption Nuclear dynamics can be approximated by classical trajectories moving on adiabatic potential surfaces, with hops to mimic nonadiabatic transitions.
Standard surface hopping assumption, used in Eqs. (8)-(9).
domain assumption The quantum momentum computed from a frozen-Gaussian reconstruction of the nuclear density provides a good approximation to the true quantum momentum.
Adopted from CT-MQC; the paper notes this form violates population conservation and requires an ad hoc correction (Section 2).
domain assumption The time-dependent scalar potential from the exact factorization is well approximated piecewise by adiabatic surfaces, so purely adiabatic forces can be used.
Based on Ref 41; used to justify Eq. (8).
ad hoc to paper Energy sharing among trajectories preserves the accuracy of the underlying quantum-classical dynamics.
The paper states the implementation of energy sharing is arbitrary and only an initial attempt (Section 2).

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Cite this review

Pith. "Pith review of A coupled-trajectory approach for decoherence, frustrated hops and internal consistency in surface hopping." pith.science (2026). https://pith.science/paper/4MMZN4S6

@misc{pith2026241204958,
  author       = {Pith},
  title        = {Pith review of: A coupled-trajectory approach for decoherence, frustrated hops and internal consistency in surface hopping},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4MMZN4S6}},
  note         = {Machine review of arXiv:2412.04958}
}

read the original abstract

We address the issues of decoherence, frustrated hops and internal consistency in surface hopping. We demonstrate that moving away from an independent-trajectory picture is the strategy which allows us to propose a robust surface hopping scheme overcoming all these issues at once. Based on the exact factorization and on the idea of coupled trajectories, we consider the swarm of trajectories, that mimics the nuclear dynamics in nonadiabatic processes, as a unique entity. In this way, imposing energy conservation of the swarm and allowing the trajectories to share energy when hops occur clearly indicates the route towards a new surface hopping scheme. Encouraging results are reported, in terms of electronic and vibrational time-dependent properties on the photodynamics of fulvene and 4-(dimethyloamino)benzonitrile, modeled with a full-dimensional linear vibronic coupling Hamiltonian.

Figures

Figures reproduced from arXiv: 2412.04958 by the authors.

**Figure 2.** Adiabatic population decays of S1 in fulvene predicted by the different surface hopping schemes discussed in the main text. The lighter lines in each panel show the dynamics with different velocity rescaling procedures and treatments of frustrated hops, while the bold darker lines show the average of the individual runs. Solid lines show the average electronic populations (P) and dashed lines the fractions of traje… view at source ↗

**Figure 3.** Spatial evolution of the swarms of trajectories along the C=CH [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Average kinetic energy of the ensemble of trajectories along the C=CH [PITH_FULL_IMAGE:figures/full_fig_p023_4.png]

**Figure 5.** Figure 5: Top panel: Population of S1 in fulvene as a function of time calculated with CCT-TSH using different flavors of the overlap-based: using the fewest-switches procedure combined with a Wigner sampling of the initial conditions (in light-brown), using the fewestswitches …

**Figure 6.** Figure 6: Adiabatic population decays of S2 in DMABN predicted by the dfferent surface hopping schemes, with TSH-ED (in green), with the equity-based scheme (in yellow), with the overlap-based scheme (in orange), with the quantum-momentum-based scheme (in red). Solid lines show …

**Figure 7.** Figure 7: Same as in Fig. 5 but for DMABN [PITH_FULL_IMAGE:figures/full_fig_p028_7.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [63]

Automatized protocol and interface to simulate QM/MM time-resolved transient absorption at TD-DFT level with COBRAMM

(7) Avagliano, D.; Bonfanti, M.; Nenov, A.; Garavelli, M. Automatized protocol and interface to simulate QM/MM time-resolved transient absorption at TD-DFT level with COBRAMM. J. Comput. Chem. 2022, 43, 1641–1655. (8) Wang, L.; Akimov, A.; Prezhdo, O. V. Recent progress in surface hopping: 2011–2015. J. Phys. Chem. Lett. 2016, 7, 2100–2112. (9) Akimov, A....

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G.; Agostini, F

(64) Eich, F. G.; Agostini, F. The adiabatic limit of the exact factorization of the electron- nuclear wave function. J. Chem. Phys. 2016, 145, 054110. (65) Agostini, F.; Abedi, A.; Gross, E. K. U. Classical nuclear motion coupled to electronic non-adiabatic transitions. J. Chem. Phys. 2014, 141, 214101. (66) Agostini, F. An exact-factorization perspectiv...

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K.; Abedi, A.; Gross, E

(67) Agostini, F.; Min, S. K.; Abedi, A.; Gross, E. K. U. Quantum-classical non-adiabatic dynamics: Coupled- vs. independent-trajectory methods. J. Chem. Theory Comput. 2016, 12, 2127–2143. (68) Talotta, F.; Morisset, S.; Rougeau, N.; Lauvergnat, D.; Agostini, F. Internal conversion and intersystem crossing with the exact factorization. J. Chem Theory Com...

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(70) Curchod, B. F. E.; Agostini, F. On the Dynamics through a Conical Intersection. J. Phys. Chem. Lett. 2017, 8, 831–837. (71) Pollien, A.; Villaseco Arribas, E.; Lauvergnat, D.; Agostini, F. Exact-factorisation study of the photochemistry of phenol. Mol. Phys. 2024, e2378960. (72) Villaseco Arribas, E.; Maitra, N. T. Energy-conserving coupled trajector...

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M.; Sanchez-Murcia, P

29 (4) Ibele, L. M.; Sanchez-Murcia, P. A.; Mai, S.; Nogueira, J. J.; Gonz´ alez, L. Excimer intermediates en route to long-lived charge-transfer states in single-stranded adenine DNA as revealed by nonadiabatic dynamics. J. Phys. Chem. Lett. 2020, 11, 7483–

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C.; Preston, R

(2) Tully, J. C.; Preston, R. K. Trajectory surface hopping approach to nonadiabatic molecular collisions: the reaction of H+ with D2. J. Chem. Phys. 1971, 55, 562–572. (3) Tully, J. C. Molecular dynamics with electronic transitions. J. Chem. Phys. 1990, 93,

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[7] [2022]

(10) Tully, J. C. Perspective: Nonadiabatic dynamics theory. J. Chem. Phys. 2012, 137 . (11) Qiu, T.; Climent, C.; Subotnik, J. E. A Practical Approach to Wave Function Prop- agation, Hopping Probabilities, and Time Steps in Surface Hopping Calculations. J. Chem. Theory Comput. 2023, 19, 2744–2757. (12) Mai, S.; Marquetand, P.; Gonz´ alez, L. Nonadiabatic...

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M.; Persico, M.; Granucci, G.; Agostini, F

(37) Pieroni, C.; Sangiogo Gil, E.; Ibele, L. M.; Persico, M.; Granucci, G.; Agostini, F. Investigating the photodynamics of trans-azobenzene with coupled trajectories. J. Chem. Theory Comput. 2024, 20, 580–596. (38) Subotnik, J. E.; Jain, A.; Landry, B.; Petit, A.; Ouyang, W.; Bellonzi, N. Understand- ing the surface hopping view of electronic transition...

work page 2024