Transition States Energies from Machine Learning: An Application to Reverse Water-Gas Shift on Single-Atom Alloys

This paper demonstrates that a machine learning model based on Gaussian process regression with a Wasserstein Weisfeiler-Lehman graph kernel significantly improves the accuracy and efficiency of predicting transition state energies and turnover frequencies for the reverse water-gas shift reaction on single-atom alloys compared to traditional scaling relations, thereby enabling robust high-throughput screening of new catalysts.

The Gist

Imagine you are a chef trying to invent the perfect new recipe for a dish that turns carbon dioxide (a greenhouse gas) into useful fuel. To do this, you need to find the right "kitchen" (a catalyst material) where the ingredients can mix and transform efficiently.

In the world of chemistry, this "kitchen" is a metal surface, and the "cooking steps" involve molecules jumping over energy hills called Transition States. The higher the hill, the harder it is to cook the dish.

Here is the problem: Finding the exact height of these energy hills using traditional computer simulations (called DFT) is like trying to measure the height of a mountain by climbing every single peak. It takes forever, costs a fortune in computer power, and is incredibly slow.

The Old Way: The "Rule of Thumb" Guess

Previously, scientists used a simple "rule of thumb" (called BEP relations) to guess the height of these hills. They assumed that if they knew how much energy it took to stick an ingredient to the pan (adsorption), they could easily guess how hard it would be to cook it (transition state).

Think of it like guessing the difficulty of a video game level just by looking at the starting screen. It works okay for simple games, but for complex ones—like the Single-Atom Alloys (SAAs) used in this study—it often fails. SAAs are like special pans where a single drop of gold is embedded in a copper pan. They behave very differently than regular pans, breaking the old rules of thumb.

The New Way: The "Smart AI Chef"

The authors of this paper built a Machine Learning (ML) model that acts like a super-smart AI chef. Instead of just using a simple rule, this AI looks at the entire "shape" of the kitchen and the ingredients.

1. The Graph Kernel (The Map): The AI doesn't just look at numbers; it draws a map (a graph) connecting every atom to its neighbors, like a subway map showing how stations are linked. It uses a special mathematical tool (Wasserstein Weisfeiler-Lehman) to compare these maps.
2. The Training: They fed the AI a massive cookbook of 1,400+ recipes (adsorbates) and 650+ cooking steps (transition states) calculated by the slow, expensive supercomputers.
3. The Prediction: Once trained, the AI can look at a new metal pan and instantly predict the energy hills with incredible accuracy, without needing to climb the mountain first.

Why This Matters: The "Turnover Frequency" (TOF)

The ultimate goal is to know how fast the catalyst works. This is called the Turnover Frequency (TOF)—basically, how many dishes the chef can serve per hour.

• The Old Method: When scientists used the simple "rule of thumb" to guess the energy hills, their prediction for how fast the chef could work was off by a huge margin—sometimes 10 times too high or too low. It's like guessing a car can drive 100 mph when it's actually stuck in traffic.
• The New Method: When they used the AI model, the prediction error dropped by almost 90%. Suddenly, the speed estimate was much closer to reality. This is crucial because if you think a catalyst is fast when it's actually slow, you waste time and money building a factory that doesn't work.

The Results: Finding the Golden Pan

Using this AI tool, the researchers screened dozens of new metal combinations. They found some exciting discoveries:

• Copper and Silver are usually slow cooks for this reaction. But, if you add a tiny bit of Nickel or Iron (creating a Single-Atom Alloy), they suddenly become very fast!
• Cobalt and Iron usually get "clogged" with oxygen (like a pan covered in burnt food), stopping the cooking. But when mixed with other metals, the clogging disappears, and they start working well.

The Big Picture

This paper is a breakthrough because it combines three powerful tools:

1. DFT: The slow, accurate "gold standard" for a few examples.
2. Machine Learning: The fast, smart "AI" that learns from the gold standard to predict the rest.
3. Microkinetic Modeling: The "business plan" that calculates how fast the whole factory will run.

By using the AI to skip the slow calculations, scientists can now screen thousands of new materials in the time it used to take to check just a handful. This accelerates the discovery of catalysts that can help us clean up our atmosphere and create sustainable fuels, turning the "impossible" task of finding the perfect catalyst into a manageable, fast process.

Technical Summary

1. Problem Statement

The rational design of heterogeneous catalysts relies heavily on accurate predictions of Transition State (TS) energies, which determine reaction rates. However, obtaining these energies via first-principles methods like Density Functional Theory (DFT) is computationally expensive, creating a bottleneck for high-throughput screening of complex materials and reaction networks.

Traditional approaches to estimate TS energies rely on:

• Brønsted-Evans-Polanyi (BEP) relations: Linear correlations between activation energy and reaction enthalpy.
• Thermochemical Scaling Relations (TSRs): Linear correlations between adsorption energies of different species.

Limitations: These linear models often fail for complex materials like Single-Atom Alloys (SAAs), where the unique electronic and geometric environments break the linear scaling rules. Furthermore, they lack the flexibility to capture non-linear relationships inherent in complex catalytic systems, leading to significant errors in predicted catalytic activity (Turnover Frequency, TOF).

2. Methodology

The authors propose a hybrid workflow combining DFT, advanced Machine Learning (ML), and microkinetic modeling.

A. Data Generation (DFT)

• System: Reverse Water-Gas Shift (RWGS) reaction ($CO_2 + H_2 \rightarrow CO + H_2O$).
• Materials: 6 pure metals (Rh, Pd, Co, Ni, Cu, Au) and 12 Single-Atom Alloys (SAAs) on (100) and (111) facets.
• Dataset: 1,448 unique adsorbate structures and 650 TS structures covering 10 intermediates and 14 elementary steps.
• Computational Details: DFT calculations performed using Quantum Espresso with the BEEF-vdW functional. TSs identified via Climbing-Image Nudged Elastic Band (CI-NEB).

B. Machine Learning Models

The study compares three modeling approaches:

1. Linear Models: Traditional TSR and BEP relations.
2. Ensemble Decision Trees (EDT): Random Forest, XGBoost, and LightGBM using averaged atomic features.
3. WWL-GPR (Proposed): A Wasserstein Weisfeiler-Lehman Graph Kernel combined with Gaussian Process Regression.
   • Graph Representation: Atomic structures are mapped to graphs where nodes are atoms and edges represent chemical bonds.
   • Features: Nodes are decorated with a rich set of descriptors (90 total), including:
     • Atomic physical constants (electronegativity, radius, etc.).
     • Electronic features from Projected Density of States (PDOS) of the clean surface (d-band center, filling, etc.).
     • Geometric features (coordination numbers).
     • Gas-phase molecule features (HOMO/LUMO).
     • SOAP descriptors (Smooth Overlap of Atomic Positions) for the surface and molecules.
   • TS Specifics: For TS structures, the graph includes the union of edges from the Initial State (IS) and Final State (FS) to account for unknown bonding configurations during the transition.

C. Validation and Uncertainty Quantification

• Cross-Validation: Stratified group K-fold cross-validation (K=6) to ensure models are tested on unseen materials.
• Ensemble Approach: To estimate uncertainty, the training data is subsampled to create an ensemble of models. The standard deviation of predictions serves as the uncertainty metric.
• Microkinetic Modeling: Predicted energies are fed into mean-field microkinetic models (using Cantera) to calculate TOFs at 500°C and 1 atm. Errors in energy predictions are propagated to TOF errors.

3. Key Contributions

1. Extension of WWL-GPR to TS Energies: The authors successfully extended a graph-based ML model, previously used for adsorption energies, to predict Transition State energies directly, bypassing the need for linear BEP approximations.
2. Superior Accuracy over Linear Models: The study demonstrates that graph-based ML significantly outperforms traditional linear scaling relations (TSR/BEP) and standard tree-based ML models (LightGBM) for both adsorption and TS energies.
3. Error Propagation Analysis: The paper quantifies how small errors in energy prediction (eV scale) propagate into large errors in catalytic activity (orders of magnitude in TOF).
4. SAA Screening: Application of the model to screen new SAA catalysts, identifying promising candidates that outperform pure metals in RWGS activity and selectivity.

4. Key Results

Energy Prediction Accuracy

• Adsorption Energies:
  • TSR (Linear): MAE = 0.212 eV.
  • LightGBM: MAE = 0.206 eV.
  • WWL-GPR: MAE = 0.149 eV (Significant improvement).
• Transition State Energies:
  • BEP (Linear): MAE = 0.200 eV.
  • LightGBM: MAE = 0.174 eV.
  • WWL-GPR: MAE = 0.154 eV (Lowest error, most localized distribution).
• Out-of-Domain Performance: When predicting energies for materials with elements not in the training set, WWL-GPR still maintained robust performance, especially when reference species (CO*, H*, O*) were explicitly calculated.

Impact on Microkinetic Modeling (TOF)

• Error Reduction: Using WWL-GPR energies reduced the Mean Absolute Error (MAE) in predicted TOF from 1.77 orders of magnitude (using linear models) to 0.97 orders of magnitude.
• Conclusion: Accurate TS energy prediction is critical; even small improvements in energy accuracy lead to nearly an order-of-magnitude improvement in activity prediction reliability.

Catalyst Screening Findings

• SAAs vs. Pure Metals:
  • Ni and Rh: High activity but suffer from low selectivity (methanation) and coking.
  • SAAs (e.g., Ni+Rh, Rh+Pt): Showed improved performance, potentially breaking scaling relations to enhance selectivity.
  • Cu and Ag: Generally low activity, but doping with Ni or Fe (e.g., Cu+Ni, Cu+Fe) significantly enhanced TOF.
  • Co, Ru, Fe: Prone to poisoning by excessive O* coverage. Doping with Pt or Rh reduced O* coverage and improved activity.
• Reaction Pathways: The model identified that while the CO-O path (direct $CO_2^*$ dissociation) is dominant for most systems, the COO-H path is preferred on Au-based SAAs. The H-COO path was relevant only for Co(111).
• Rate-Determining Steps (RDS): For most systems, $CO_2^*$ dissociation is the RDS. However, for SAAs favoring the COO-H path, the RDS shifts to $CO_2^*$ hydrogenation. Notably, $H_2O^*$ formation was identified as the RDS for a broad range of SAAs, a step where SAAs break the BEP relationship of pure metals.

5. Significance

• Robust Framework: The work establishes a robust framework for catalyst screening that combines DFT, graph-based ML, and microkinetic modeling. It moves beyond the limitations of linear scaling relations, which are increasingly inadequate for complex materials like SAAs.
• Accelerated Discovery: By reducing the computational cost of TS searches while maintaining high accuracy, this approach enables the rapid screening of vast chemical spaces for complex reactions like RWGS.
• Design Principles: The study provides specific design rules for RWGS catalysts, highlighting the potential of earth-abundant SAAs (Cu/Ag doped with Ni/Fe) as economically viable alternatives to noble metals.
• Methodological Advancement: It validates the use of Wasserstein Weisfeiler-Lehman graph kernels for capturing the complex, non-linear energetic landscapes of transition states, a capability essential for next-generation catalyst design.