# Challenges for Kinetics Predictions via Neural Network Potentials: A Wilkinson’s Catalyst Case

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Reaction Path Search Using the AFIR Method

#### 2.2. Dataset Description

_{3})

_{3}, the simplified catalyst RhCl(PH

_{3})

_{3}has been considered.

#### 2.3. GTM Visualization

#### 2.4. 3D Pairwise-Sorted Distance-Based Descriptors

_{p}and ${X}_{ip}$ are the p-th term of the descriptor characterizing, respectively, the entire subset and the i-th structure, i iterates over 3D structures belonging to the same 2D structure, ${w}_{i}$ is the weight associated to the structure i, ${E}_{i}$ is the relative potential energy of i compared to the lowest energy observed within the reaction path network, ${k}_{B}$ is the Boltzmann constant, and $T=300\mathrm{K}$ is the considered temperature.

#### 2.5. Neural Network Potential Architecture

#### 2.6. NNP(+xTB) Models

## 3. Results

#### 3.1. Reaction Path Network for Hydrogenation Using a Simplified Wilkinson’s Catalyst

_{2}, catalyzed by the Wilkinson’s catalyst (i.e., RhCl(PPh

_{3})

_{3}), involves the following steps after the PPh

_{3}dissociation: oxidative addition of H

_{2}to the metal complex; alkene coordination; alkene insertion; and reductive elimination of alkane [68].

_{3})

_{2}, and reported that the oxidative addition of H

_{2}occurs before the ethylene coordination. In the present study, we have considered the non-dissociated simplified catalyst RhCl(PH

_{3})

_{3}by explicitly modeling all three PH

_{3}ligands. For this system, we found that the ethylene coordination is the first step of the hydrogenation, producing RhCl(PH

_{3})

_{3}(C

_{2}H

_{4}); followed by the dissociation of a PH

_{3}ligand; then, the oxidative addition of H

_{2}, ethylene insertion, and reductive elimination of ethane proceed with two PH

_{3}ligands. Finally, the initial RhCl(PH

_{3})

_{3}catalyst is restored, completing the catalytic cycle.

_{3})

_{2}(C

_{2}H

_{4}), which is not active enough to react with H

_{2}at 1 atmosphere, likely due to the large π-acidity of the ethylene ligand [42]. This observation seems consistent with the preferential initial coordination of ethylene in the present study, even though further research is required to understand if the mismatch on the number of coordinated phosphines (2 × PPh

_{3}vs. 3 × PH

_{3}) is solely due to steric effects. Unlike Wilkinson’s catalyst, we found that the hydrogenation of ethylene can proceed with the simplified catalyst. This outcome could be partially due to a lower (compared to the original PPh

_{3}) simulated π-acidity of the PH

_{3}ligands [70], especially since this property was found to be particularly sensitive to the accuracy of the P 3d orbitals description [71] (see Supplementary Materials Section S8 for an in-depth analysis). However, such analysis is out of the scope of the present study. In general, for practical machine learning applications, the quality of the dataset is essential. To this end, it is desirable to find the most adapted level of theory by performing comparison against higher precision calculations, such as CCSD(T)-F12 [72,73]. One should note that our method is a priori, compatible with any level of theory.

#### 3.2. Data Visualization with GTM

_{3}groups; whereas, in high-energy areas d and e, one of the PH

_{3}groups is oriented toward the rhodium by its hydrogen atoms. The latter areas correspond to some sort of dead-end of the reaction network. Class landscapes demonstrating distribution of 3D structures of the product and those formed in main reaction steps 1–6 are given in Supplementary Materials Figure S11. Notice that, thanks to the Boltzmann-like weighting of descriptors, the representative projection of each group is always located near its lowest-energy cluster.

#### 3.3. Applicability of Neural Network Potentials to AFIR-Based Reaction Path Search

#### 3.3.1. NNP Performance on Pre-Obtained Geometries

#### 3.3.2. Reaction Path Search Using the NNP Model

- Lack of physics: While general-purpose NNPs, such as SpookyNet, do respect fundamental symmetries (translation, rotation, …), their functional forms (i.e., the mathematical models) are not physics-based. In particular, their asymptotic behavior is not governed by physical principles. Although SpookyNet models already include additional trainable terms that are physics-inspired (E
_{ZBL}, E_{D}_{4}and E_{elec}), these terms do not seem sufficient to ensure physical asymptotic behavior outside the training domain. - Training bias: Due to the aforementioned lack of physics, the NNP considerably relies on the training data, yet the dataset does not contain strongly broken geometries. Indeed, such geometries are not encountered during the DFT-based search, because all paths leading to them would be rightfully assessed as too high in energy for the exploration to continue. Therefore, the trained NNPs cannot properly handle these extreme geometries, leading them to be poorly described.
- Strong exploration forces: Even if sufficient training data is available in the accessible valleys of a potential energy surface (i.e., chemically reasonable geometries), we believe that applying a strong external force can drive a properly described system outside the locally well-defined valleys of the fitted potential.

#### 3.3.3. Δ-Learning Solution for Robust NNP-Based Models

- Strong exploration forces are a powerful tool to efficiently sample rare events [76], so we believe that one should focus on designing models that can support them, instead of removing them.
- SpookyNet models need to be trained on broken geometries to properly describe them. We argue that complementing the training dataset a priori with broken geometries is not reasonable, because one cannot easily predict in advance the pitfalls of a fitted potential, and one cannot reasonably include all possible broken geometries in the training set. A simple argument to convince the reader is to consider N atoms randomly distributed in a box: the probability that the resulting geometry is chemically reasonable is close to zero, therefore illustrating the inconceivably large ratio of broken geometries over reasonable geometries. We further argue that such training bias toward reasonable geometries in available datasets is actually desirable, because we believe it is unreasonable to waste computational resources on unreasonable geometries.

**Hypothesis 1 (H1).**

**Hypothesis 2 (H2).**

_{ZBL}, E

_{D4}and E

_{elec}).

#### 3.3.4. Kinetic Study from Reaction Path Search Using NNP(+xTB)

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Description of the AFIR method. An artificial force is applied to easily cross reaction barriers.

**Figure 3.**Overview of the novel approach described in this article: using NNP-based models trained on prior ab initio data to support AFIR-based reaction path searches.

**Figure 4.**Reaction scheme of hydrogenation using a simplified catalyst, inspired by Wilkinson’s catalyst (i.e., RhCl(PPh

_{3})

_{3}).

**Figure 6.**Reaction path network obtained with the AFIR method and kinetic-based navigation, computed at the DFT level. Boxes represent equilibrium structures; edges represent reaction paths. The main reaction steps (depicted in dark grey) correspond to intermediate structures #1–6.

**Figure 7.**GTM energy landscape for the entire DFT-based network, with projected groups. Groups belonging to the proposed hydrogenation reaction path are labeled, where numbers correspond to the reaction intermediates involved, in order. The corresponding reaction path is highlighted by red arrows.

**Figure 8.**GTM class landscape showing distribution of 3D structures corresponding to reactants. 5 different clusters (labelled with letters) can be identified and correspond to distinct conformations.

**Figure 9.**GTM landscapes describing first 20%, 50%, 80%, and 100% of the network exploration discovered in the DFT-based search. Each next map visualizes a class landscape, where the brown color corresponds to the zones populated exclusively by “new” (with respect to the previous map) structures, and the blue color—to the zones populated by “old” structures. Notice that the map accommodating the first 20% contains only “new” structures.

**Figure 10.**Dataset splitting scheme into training set, validation set, and test set. First, a search-related timestamp is chosen (e.g., when 50% of the network’s paths has been explored by the search, which is equivalent to: when the search is half-completed). The geometries corresponding to paths already explored before this timestamp are grouped into the train/validation set, and the test set is composed of all geometries corresponding to paths that were not yet discovered at this time of the search. The train/validation set is then split into a training set and a validation set randomly, while ensuring that all geometries corresponding to a single path are either within the training set or the validation set (i.e., validation geometries correspond to paths which are not covered in the training set, except for the EQs shared with training paths).

**Figure 11.**Performance of SpookyNet models trained on the first paths explored during the DFT-powered AFIR-based reaction path search. The energy predictions and energy references (i.e., DFT energies) are displayed for the remaining geometries, with transparency for better readability. (

**a**) Model trained on the first 20% of paths explored, the remaining 80% are represented, R

^{2}= 0.97; (

**b**) Model trained on the first 50% of paths explored, the remaining 50% are represented, R

^{2}= 0.995; (

**c**) Model trained on the first 80% of paths explored, the remaining 20% are represented, R

^{2}= 0.998.

**Figure 12.**Performance of SpookyNet when powering a local AFIR-based exploration around the most stable reactants conformer. Each point represents a PES stationary point geometry (i.e., an approximate TS or EQ) obtained during the search. The energy predictions are generated during the search, and the energy references (i.e., DFT energies) are computed a posteriori. Here, the largest errors were found on structures with no apparent steric clashes, but with dissociated structures and/or isolated atoms. The model was trained on the first 80% of paths explored during the preliminary DFT-powered search, R

^{2}= −76.

**Figure 13.**Principle behind the Δ-learning solution as a physics-based safeguard. The physics-based term included via Δ-learning serves as a safeguard to prevent reaching broken geometries due to the AFIR force.

**Figure 14.**Performance of NNP(+xTB) models when powering a local AFIR-based exploration around the most stable reactants conformer. Each point represents a PES stationary point geometry (i.e., an approximate TS or EQ) obtained during the search. The energy predictions are generated during the search, and the energy references (i.e., DFT energies) are computed a posteriori. xTB and NNP(+xTB) predictions for the same geometry are connected by a line: a red line, if xTB only is closer to DFT, and a green line, if the NNP contribution is beneficial. The energies potentials are shifted to match each other on the WilkinsonAFIRdb dataset. (

**a**) Model was trained on the first 20% of paths explored during the preliminary DFT-powered search, R

^{2}= 0.74; (

**b**) Model trained on the first 50% of paths explored, R

^{2}= 0.79; (

**c**) Model trained on the first 80% of paths explored, R

^{2}= 0.93.

**Figure 15.**Performance of GFN2-xTB when powering a local AFIR-based exploration around the most stable reactants conformer. Each point represents a PES stationary point geometry (i.e., an approximate TS or EQ) obtained during the search. The xTB energies are generated during the search, and the energy references (i.e., DFT energies) are computed a posteriori. As always, xTB energies are shifted by the exact same amount that was used to minimize the Mean Square Error on the WilkinsonAFIRdb dataset, R

^{2}= −6.

**Table 1.**Predicted yields at different temperatures, from AFIR-based reaction path search using different potentials.

Predicted Yield | GFN2-xTB | NNP(+xTB) 20% Training | NNP(+xTB) 50% Training | NNP(+xTB) 80% Training | DFT |
---|---|---|---|---|---|

250 K | 0.50% | 0.00% | 2.09% | 31.42% | 98.47% |

300 K | 1.42% | 0.00% | 96.47% | 100% | 100% |

350 K | 2.79% | 0.00% | 99.95% | 99.98% | 100% |

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**MDPI and ACS Style**

Staub, R.; Gantzer, P.; Harabuchi, Y.; Maeda, S.; Varnek, A.
Challenges for Kinetics Predictions via Neural Network Potentials: A Wilkinson’s Catalyst Case. *Molecules* **2023**, *28*, 4477.
https://doi.org/10.3390/molecules28114477

**AMA Style**

Staub R, Gantzer P, Harabuchi Y, Maeda S, Varnek A.
Challenges for Kinetics Predictions via Neural Network Potentials: A Wilkinson’s Catalyst Case. *Molecules*. 2023; 28(11):4477.
https://doi.org/10.3390/molecules28114477

**Chicago/Turabian Style**

Staub, Ruben, Philippe Gantzer, Yu Harabuchi, Satoshi Maeda, and Alexandre Varnek.
2023. "Challenges for Kinetics Predictions via Neural Network Potentials: A Wilkinson’s Catalyst Case" *Molecules* 28, no. 11: 4477.
https://doi.org/10.3390/molecules28114477