Markov State Models for Tracking Reaction Dynamics on Catalytic Nanoparticles
Pith reviewed 2026-05-16 02:28 UTC · model grok-4.3
The pith
Nanoparticle corners and edges slow hydrogen association and dissociation on rhodium, producing non-monotonic rates with concentration unlike standard transition state theory.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Markov state models constructed from molecular dynamics simulations of hydrogen on rhodium catalysts demonstrate that nanoparticle features such as corners and edges slow the association and dissociation processes, while cooperative hydrogen-hydrogen interactions produce a non-monotonic dependence of the rates on surface concentration that standard transition state theory does not predict.
What carries the argument
Markov state models built directly from molecular dynamics trajectories generated with machine-learned interatomic potentials, which coarse-grain the complex surface dynamics into discrete kinetic states while retaining effects from structural fluctuations and adsorbate cooperativity.
If this is right
- Nanoparticle geometries produce slower association and dissociation rates than flat slabs because of edge and corner effects.
- Rates vary non-monotonically with increasing hydrogen coverage due to cooperative adsorbate interactions.
- Standard transition state theory misses the observed rate behavior in systems with fluctuating surfaces and multiple interacting species.
- The Markov state model framework enables extraction of interpretable kinetics from direct simulations of complex catalytic environments.
Where Pith is reading between the lines
- Catalyst design could target specific nanoparticle shapes to modulate rates through these structural slowing effects.
- The same cooperative non-monotonic behavior may appear in other multi-adsorbate reactions on nanoparticles.
- The approach opens a route to predict long-time kinetics for nanoparticles too large for exhaustive direct simulation.
- Validation on experimental rhodium nanoparticle samples at controlled coverages would test the predicted rate trends.
Load-bearing premise
The finite molecular dynamics trajectories produce Markov state models that faithfully recover the true long-time kinetics without significant non-Markovian effects or sampling errors, and the machine-learned potentials reproduce the correct barriers and dynamics for the rhodium-hydrogen system.
What would settle it
Experimental measurements of hydrogen association and dissociation rates on rhodium nanoparticles across a range of coverages that fail to show slower kinetics on nanoparticles or the predicted non-monotonic concentration dependence would falsify the claims.
read the original abstract
Markov state models (MSMs) are a powerful tool to analyze and coarse-grain complex dynamical data into interpretable kinetic processes. This capability is particularly important in heterogeneous catalysis, where a medley of reactants and intermediates interact on surfaces that might simultaneously experience structural fluctuations. For these very complex systems, standard transition state theory (TST) approaches are no longer appropriate, motivating alternative approaches that can retain dynamical complexity while providing physical insight. With machine learned interatomic potentials being more and more ubiquitous, directly simulating complex catalytic systems with molecular dynamics (MD) is becoming increasingly feasible. Extending MSMs to dynamically coarse grain MD simulation data of catalytic processes, we analyze hydrogen dynamics on rhodium catalysts with slab and nanoparticle geometries over a range of hydrogen surface concentrations. Somewhat counterintuitively, nanoparticle features, such as corners and edges, effectively slow down the association/dissociation process, and the cooperative behavior of hydrogen-hydrogen interactions leads to a non-monotonic concentration dependence of the rates, which would not be predicted with standard TST.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript constructs Markov state models (MSMs) from molecular dynamics trajectories generated with machine-learned interatomic potentials to coarse-grain hydrogen association and dissociation kinetics on rhodium slab and nanoparticle surfaces across a range of surface coverages. The central claims are that nanoparticle-specific features (corners and edges) slow the association/dissociation rates relative to flat slabs and that cooperative H-H interactions produce a non-monotonic dependence of the rates on hydrogen concentration, an effect not captured by standard transition-state theory.
Significance. If the MSM-derived rates are shown to be converged and free of significant sampling or non-Markovian artifacts, the work would provide a concrete demonstration that dynamical coarse-graining can reveal coverage-dependent and morphology-dependent effects in heterogeneous catalysis that are inaccessible to TST, with direct implications for interpreting and designing nanoparticle catalysts.
major comments (3)
- [MSM construction and validation] The manuscript provides no Chapman-Kolmogorov tests, implied-timescale convergence plots versus lag time, or direct comparison of MSM rates to raw MD transition counts. These checks are required to establish that the reported non-monotonic concentration dependence and nanoparticle slowing are not artifacts of lag-time choice or undersampling of rare corner/edge transitions.
- [Results on concentration dependence] The claim that the observed non-monotonic rates 'would not be predicted with standard TST' is stated without an explicit TST calculation (e.g., via harmonic TST or committor analysis) performed on the same ML potential and surface models; a side-by-side comparison is needed to make the contrast quantitative.
- [Rate extraction and statistics] No error bars, bootstrap estimates, or convergence tests with respect to total MD sampling time are reported for the extracted rates, despite the acknowledged rarity of association/dissociation events on heterogeneous nanoparticles.
minor comments (2)
- [Methods] Notation for state definitions and the precise definition of 'association' versus 'dissociation' events should be clarified in the methods to allow reproducibility.
- [Figures] Figure captions should explicitly state the lag time used for each MSM and the number of independent trajectories contributing to each coverage.
Simulated Author's Rebuttal
We thank the referee for their detailed and constructive review. We have carefully addressed each of the major comments by adding the requested validations, comparisons, and statistical analyses to the revised manuscript.
read point-by-point responses
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Referee: The manuscript provides no Chapman-Kolmogorov tests, implied-timescale convergence plots versus lag time, or direct comparison of MSM rates to raw MD transition counts. These checks are required to establish that the reported non-monotonic concentration dependence and nanoparticle slowing are not artifacts of lag-time choice or undersampling of rare corner/edge transitions.
Authors: We agree that these standard MSM validation tests are essential. We have performed Chapman-Kolmogorov tests at multiple lag times and included the corresponding plots in the revised Supplementary Information. Implied timescale convergence plots versus lag time are also added, demonstrating convergence for the chosen lag time. Additionally, we now provide a direct comparison between the MSM-derived rates and the raw transition counts from the MD trajectories, confirming consistency. These additions address concerns about potential artifacts. revision: yes
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Referee: The claim that the observed non-monotonic rates 'would not be predicted with standard TST' is stated without an explicit TST calculation (e.g., via harmonic TST or committor analysis) performed on the same ML potential and surface models; a side-by-side comparison is needed to make the contrast quantitative.
Authors: We acknowledge that an explicit comparison to TST would strengthen the claim. In the revised manuscript, we have computed harmonic TST rates using the same machine-learned potential for the slab and nanoparticle models at various coverages. The results are compared side-by-side with the MSM rates in a new figure, showing that TST predicts a monotonic decrease with coverage, while MSM captures the non-monotonic behavior due to cooperative effects. This quantitative contrast is now included in the main text. revision: yes
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Referee: No error bars, bootstrap estimates, or convergence tests with respect to total MD sampling time are reported for the extracted rates, despite the acknowledged rarity of association/dissociation events on heterogeneous nanoparticles.
Authors: We agree that statistical uncertainties should be reported. We have now performed bootstrap resampling on the MD trajectories to estimate error bars on the rates, which are included in the revised figures. Additionally, we have conducted convergence tests by varying the total sampling time and shown that the rates stabilize beyond a certain threshold, with these results added to the Supplementary Information. revision: yes
Circularity Check
No circularity: rates extracted directly from MD trajectories via MSMs
full rationale
The paper builds Markov state models from finite MD trajectories of hydrogen on Rh surfaces and nanoparticles, then reports association/dissociation rates and concentration dependence as direct outputs of the estimated transition matrices. No equations define a quantity in terms of itself, no fitted parameter is relabeled as an independent prediction, and no load-bearing premise reduces to a self-citation chain. The non-monotonic behavior and nanoparticle slowing are presented as empirical findings from the data-driven MSM, not as derivations that are tautological with their inputs. The derivation chain is therefore self-contained against external benchmarks.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption The dynamics of hydrogen on Rh surfaces can be coarse-grained into a Markovian state model without significant memory effects
discussion (0)
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