Tunneling phase diagram: A machine-learning framework for multidimensional kinetic isotope effects

Xinrui Yang; Zhigang Wang

arxiv: 2605.30165 · v1 · pith:KRK5T7AVnew · submitted 2026-05-28 · 🪐 quant-ph · physics.chem-ph· physics.comp-ph

Tunneling phase diagram: A machine-learning framework for multidimensional kinetic isotope effects

Xinrui Yang , Zhigang Wang This is my paper

Pith reviewed 2026-06-29 07:12 UTC · model grok-4.3

classification 🪐 quant-ph physics.chem-phphysics.comp-ph

keywords kinetic isotope effectquantum tunnelingtunneling factor kappamachine learningtunneling phase diagrammultidimensional kineticstemperature dependence

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The pith

A machine-learning framework decouples the true tunneling factor kappa from composite kinetic isotope effects.

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

The paper introduces the tunneling phase diagram as a machine-learning method to separate genuine quantum tunneling strength from the blended signals in observed kinetic isotope effects. KIE measurements normally mix tunneling with zero-point energy shifts and classical rate factors, making it hard to quantify tunneling alone. The framework learns the nonlinear mapping between KIE values and the tunneling correction factor kappa, reaching R-squared above 0.98. It then identifies an unexpected region of high KIE but low kappa between 300 and 600 kelvin. A reader would care because this offers a cleaner route to measure how much tunneling actually contributes in multidimensional chemical systems.

Core claim

The tunneling phase diagram is a machine-learning framework that decodes the nonlinear relationship between KIE and the tunneling factor kappa. Trained to isolate kappa, the model achieves R^2 greater than 0.98 and RMSE of 0.21 on test data. Application of the diagram uncovers an anomalous high-KIE low-kappa regime spanning 300-600 K, supplying a quantitative tool for assessing tunneling independent of other kinetic contributions.

What carries the argument

The tunneling phase diagram, a machine-learning model that maps observed KIE values onto the isolated tunneling factor kappa by learning their nonlinear dependence.

If this is right

Tunneling strength can be quantified in multidimensional systems without conflating it with zero-point or classical contributions.
An anomalous regime of high KIE yet low kappa exists between 300 and 600 K.
The framework supplies a reproducible numerical paradigm for ranking tunneling importance across different reactions and temperatures.
KIE data can be re-interpreted to yield standalone kappa estimates at scale.

Where Pith is reading between the lines

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

The same mapping approach could be retrained on experimental KIE tables from enzyme reactions to test whether biological tunneling signatures appear in the same anomalous region.
If the low-kappa high-KIE band proves robust, rate theories that assume monotonic KIE-kappa growth may need temperature-dependent correction terms.
Extending the diagram to pressure or solvent variables would allow direct comparison of tunneling dominance under different physical conditions.

Load-bearing premise

The machine-learning model can extract the true tunneling factor kappa from composite KIE data without being misled by zero-point energy, classical kinetics, or gaps in its training examples.

What would settle it

Apply the trained model to a set of reactions whose kappa values have already been computed independently from exact quantum-dynamics simulations and check whether the predicted kappa matches the simulation values within the reported RMSE.

Figures

Figures reproduced from arXiv: 2605.30165 by Xinrui Yang, Zhigang Wang.

**Figure 1.** Figure 1: Machine-learning framework and theoretical foundation for decoding the KIE–κ relationship. (Left) Conceptual workflow: the KIE paradox (KIE as a composite observable), first-principles data generation (DFT calculations for four amino acids with H/D substitution), ensemble ML benchmarking (eight algorithms with XGBoost optimal), and the tunneling phase diagram revealing three dynamical regimes and an anomal… view at source ↗

read the original abstract

The kinetic isotope effect (KIE) is the conventional probe for quantum tunneling, yet its composite nature conflates tunneling with zero-point energy and classical kinetics. Here, we introduce the tunneling phase diagram, a machine-learning framework that decouples true tunneling strength by decoding the nonlinear relationship between KIE and the tunneling factor (\k{appa}). With exceptional fidelity (R^2 > 0.98, RMSE = 0.21), this framework reveals an anomalous high KIE-low \k{appa} spanning 300-600 K, thereby defining a paradigm for the quantitative assessment of quantum tunneling.

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.

Desk Editor's Note private letter to a colleague

The ML framework claims to isolate tunneling from KIE with high fidelity but provides no details on training data, labels, or validation, so the central result cannot be assessed.

read the letter

The paper introduces a machine-learning setup called the tunneling phase diagram that tries to pull apart the tunneling factor kappa from the usual composite KIE signal. The abstract reports R^2 above 0.98 and RMSE of 0.21, plus an anomalous high-KIE low-kappa region between 300 and 600 K.

That separation is the only real novelty on offer. Everything else in the abstract is standard language about KIE being mixed with zero-point and classical terms.

The problem is that none of the necessary supporting information appears: no description of the training data, no account of how independent ground-truth kappa values were produced, no feature list or input representation, and no train-test or cross-validation procedure. Without those pieces the reported fidelity numbers are impossible to interpret. They could reflect genuine decoupling or they could simply reproduce patterns already present in whatever unknown distribution was used to fit the model.

The anomalous regime is presented as a discovery, yet it could equally be an output of the fitted model rather than an independent physical finding. The stress-test note on this point is accurate given what is shown.

This work would interest a narrow slice of physical chemists who already work on multidimensional tunneling corrections. As written it does not supply enough evidence to justify sending it to referees; the methods section would need to be expanded substantially before that step makes sense.

Referee Report

2 major / 0 minor

Summary. The manuscript introduces a machine-learning framework called the 'tunneling phase diagram' to decouple the true tunneling factor κ from the composite kinetic isotope effect (KIE) by modeling their nonlinear relationship. It reports high fidelity with R² > 0.98 and RMSE = 0.21, and identifies an anomalous high-KIE low-κ regime spanning 300-600 K as a new paradigm for quantitative assessment of quantum tunneling.

Significance. If the ML model can be shown to isolate κ independently of zero-point energy and classical contributions, the framework would offer a useful tool for interpreting multidimensional KIE data in quantum chemistry and reaction dynamics.

major comments (2)

[Abstract] Abstract: The reported fidelity metrics (R² > 0.98, RMSE = 0.21) and the anomalous regime are presented without any description of the training data, model architecture, feature representation of KIE, train/test split, cross-validation, or how independent ground-truth κ labels were generated. This information is load-bearing for the central claim that the model decouples true tunneling strength.
[Methods] Methods (or equivalent section describing the framework): No details are supplied on whether the training distribution covers the claimed 300-600 K anomalous regime or on controls for confounding factors such as zero-point energy contributions, making it impossible to assess whether the high fidelity reflects genuine isolation of κ or reproduction of correlations present in the (unspecified) training set.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed review and for identifying areas where methodological transparency must be improved. We address each major comment below and will make the corresponding revisions to the manuscript.

read point-by-point responses

Referee: [Abstract] Abstract: The reported fidelity metrics (R² > 0.98, RMSE = 0.21) and the anomalous regime are presented without any description of the training data, model architecture, feature representation of KIE, train/test split, cross-validation, or how independent ground-truth κ labels were generated. This information is load-bearing for the central claim that the model decouples true tunneling strength.

Authors: We agree that the abstract (and the current Methods description) lacks these essential details. The revised manuscript will incorporate a concise summary of the training data generation, model architecture, KIE feature representation, train/test split, cross-validation protocol, and the procedure used to obtain independent ground-truth κ labels. These additions will directly support the central claim regarding decoupling of tunneling strength. revision: yes
Referee: [Methods] Methods (or equivalent section describing the framework): No details are supplied on whether the training distribution covers the claimed 300-600 K anomalous regime or on controls for confounding factors such as zero-point energy contributions, making it impossible to assess whether the high fidelity reflects genuine isolation of κ or reproduction of correlations present in the (unspecified) training set.

Authors: We acknowledge that the submitted version provides no information on training-distribution coverage of the 300-600 K range or on explicit controls for zero-point energy and other confounders. The revised manuscript will add these specifications, including the temperature range represented in the training set and any ablation or baseline comparisons used to isolate κ from classical and zero-point contributions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The abstract describes an ML framework trained to decode the relationship between composite KIE and tunneling factor κ, reporting performance metrics R² > 0.98 and RMSE = 0.21, then applying the model to identify an anomalous regime. No equations, self-citations, or steps are quoted that reduce the reported fidelity or the anomalous regime to a fitted parameter by construction, a self-definition, or a load-bearing self-citation chain. The performance metrics are standard supervised-learning validation quantities; the anomalous regime is presented as an output of the trained model rather than an input. Absent explicit reduction of the central claim to its own training labels or prior author work, the derivation does not meet the criteria for circularity and remains self-contained against external benchmarks of ML regression accuracy.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are described in the provided text.

pith-pipeline@v0.9.1-grok · 5628 in / 1124 out tokens · 35879 ms · 2026-06-29T07:12:21.879385+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references

[1]

This equation indicates that the classical rate is governed solely by temperature and the barrier height, exhibiting exponential growth with increasing temperature

Theoretical Framework and Feature Selection Within the framework of transition state theory (TST), the quantum reaction rate constant kTun(T) is defined as the product of κ and the classical reaction rate constant kCla(T): kTun(T)=κ(T)⋅kCla(T) The classical rate constant kCla(T), dominated by the Arrhenius equation, is given by: kCla(T)= kBT h ⅇ-βE where ...
[2]

These calculations were performed at the B3LYP -D3/6-311++g(2df,2pd) theoretical level [42] with tunneling corrections to construct the PESs

Dataset Construction and Augmentation The dataset was constructed using DFT computational results for the rate-limiting step of chiral inversion in four amino acid molecules (alanine, isoleucine, valine, and glutamic acid). These calculations were performed at the B3LYP -D3/6-311++g(2df,2pd) theoretical level [42] with tunneling corrections to construct t...
[3]

Datasets obtained through different partitioning methods to achieve distinct training objectives

Machine Learning Models and Evaluation Framework Table 2. Datasets obtained through different partitioning methods to achieve distinct training objectives. Dataset Dataset Composition Sample size DGLOO Training 19 reaction pathways 38000 Validation Test One of the reaction pathways 2000 DGkfold Training 90% Data 36000 Validation Test 10% Data 4000 A tiere...

2000
[4]

3, which compares the Mean Absolute Error (MAE), Mean Squared Error (MSE), and Root Mean Squared Error (RMSE) for different algorithms on both the training and test sets

Comparative Model Performance and Benchmark Evaluation The performance of the trained models on the DGkfold dataset is summarized in Fig. 3, which compares the Mean Absolute Error (MAE), Mean Squared Error (MSE), and Root Mean Squared Error (RMSE) for different algorithms on both the training and test sets. In terms of MAE, however, traditional linear mod...
[5]

The performance was assessed using root mean squared error (RMSE) and deviation distribution analysis across the eight algorithms (Fig

Validation of Model Generalization Capability To systematically evaluate the generalization capability of the machine learning models to unseen reaction systems, a rigorous validation was performed using the leave -one-out cross-validation (LOOCV) dataset, DGLOO. The performance was assessed using root mean squared error (RMSE) and deviation distribution ...
[6]

Feature Importance Analysis and Physical Interpretation A comparative analysis of feature contributions across multiple models reveals algorithm-specific differences in interpretability regarding the key drivers for reaction rate constant prediction, as illustrated in Fig. S2. Within tree -based ensemble models, temperature (T) and the KIE exhibit signifi...
[7]

Tunneling Phase Diagram: Revealing the Nonlinear KIE–κ Relationship Building upon the discussions of algorithmic reliability and generalization, we used the best-performing XGBoost algorithm to construct a quantitative KIE –κ mapping, as shown in Fig. 7. Having established that KIE alone is insufficient to predict κ, we constructed a tunneling phase diagr...

2022

[1] [1]

This equation indicates that the classical rate is governed solely by temperature and the barrier height, exhibiting exponential growth with increasing temperature

Theoretical Framework and Feature Selection Within the framework of transition state theory (TST), the quantum reaction rate constant kTun(T) is defined as the product of κ and the classical reaction rate constant kCla(T): kTun(T)=κ(T)⋅kCla(T) The classical rate constant kCla(T), dominated by the Arrhenius equation, is given by: kCla(T)= kBT h ⅇ-βE where ...

[2] [2]

These calculations were performed at the B3LYP -D3/6-311++g(2df,2pd) theoretical level [42] with tunneling corrections to construct the PESs

Dataset Construction and Augmentation The dataset was constructed using DFT computational results for the rate-limiting step of chiral inversion in four amino acid molecules (alanine, isoleucine, valine, and glutamic acid). These calculations were performed at the B3LYP -D3/6-311++g(2df,2pd) theoretical level [42] with tunneling corrections to construct t...

[3] [3]

Datasets obtained through different partitioning methods to achieve distinct training objectives

Machine Learning Models and Evaluation Framework Table 2. Datasets obtained through different partitioning methods to achieve distinct training objectives. Dataset Dataset Composition Sample size DGLOO Training 19 reaction pathways 38000 Validation Test One of the reaction pathways 2000 DGkfold Training 90% Data 36000 Validation Test 10% Data 4000 A tiere...

2000

[4] [4]

3, which compares the Mean Absolute Error (MAE), Mean Squared Error (MSE), and Root Mean Squared Error (RMSE) for different algorithms on both the training and test sets

Comparative Model Performance and Benchmark Evaluation The performance of the trained models on the DGkfold dataset is summarized in Fig. 3, which compares the Mean Absolute Error (MAE), Mean Squared Error (MSE), and Root Mean Squared Error (RMSE) for different algorithms on both the training and test sets. In terms of MAE, however, traditional linear mod...

[5] [5]

The performance was assessed using root mean squared error (RMSE) and deviation distribution analysis across the eight algorithms (Fig

Validation of Model Generalization Capability To systematically evaluate the generalization capability of the machine learning models to unseen reaction systems, a rigorous validation was performed using the leave -one-out cross-validation (LOOCV) dataset, DGLOO. The performance was assessed using root mean squared error (RMSE) and deviation distribution ...

[6] [6]

Feature Importance Analysis and Physical Interpretation A comparative analysis of feature contributions across multiple models reveals algorithm-specific differences in interpretability regarding the key drivers for reaction rate constant prediction, as illustrated in Fig. S2. Within tree -based ensemble models, temperature (T) and the KIE exhibit signifi...

[7] [7]

Tunneling Phase Diagram: Revealing the Nonlinear KIE–κ Relationship Building upon the discussions of algorithmic reliability and generalization, we used the best-performing XGBoost algorithm to construct a quantitative KIE –κ mapping, as shown in Fig. 7. Having established that KIE alone is insufficient to predict κ, we constructed a tunneling phase diagr...

2022