Transferable, Living Data Sets for Predicting Global Minimum Energy Nanocluster Geometries

Modeling of nanocluster geometries is essential for studying the dependence of catalytic activity on the available active sites. In heterogeneous catalysis, the interfacial interaction of the support with the metal can result in modification of the structural and electronic properties of the clusters. To tackle the study of a diverse array of cluster shapes, data-driven methodologies are essential to circumvent prohibitive computational costs. At their core, these methods require large data sets in order to achieve the necessary accuracy to drive structural exploration. Given the similarity in binding character of the transition metals, cluster shapes encountered for various systems show a large amount of overlap. This overlap has been utilized to construct a living data set which may be carried over across multiple studies. Iterative refinement of this data set provides a low-cost pathway for initialization of cluster studies. It is shown that utilization of transferable structural information can reduce model construction costs by more than 90%. The benefits of this approach are particularly notable for alloy systems, which possess significantly larger configurational spaces compared to the pure-phase counterparts.

1 Computational Details

ANN Construction
A basis set of radial (R), angular (A) and electronic (B) sampling functions is used to convert the coordinates of the structures in the dataset into feature vectors that are used as input for the ANN.
Evaluated here for atom i. Interactions sum over all neighbours within a cut-off radius r c .r ij denotes the distance between atom i and neighbour j. ε denote scalar atomic parameters.
The basis set for the ANN was optimised using an iterative CSR method. 1 A cut-off radius of 6.5 Å includes interactions with the third coordination shell for the metal atoms.The set of radial decay parameters η included 0, 0.025, 0.05, 0.15, 0.25, 0.5 and 1.0 Å−2 .The angular decay parameters ζ included 1, 1.5, 3, 6, 16 and 32.Two phase offsets λ = 0, 1 were included for the angular sampling.This yields 91 structural basis functions in total.The core charge, valence charge and magnetic moment were included as atomic parameters for the electronic basis set.
The ANN coefficients were optimised using the RMSprop optimiser coupled with a single-variable descent stage.The initial coefficients were obtained from the Glorot uniform initialiser. 2Gradients were calculated using mini-batches of 50 samples.A training set containing 80% of samples was used for the conjugate gradient evaluations.10% of samples was used as a validation set to evaluate convergence of the optimiser while the remaining 10% was used as a test set to determine subset biasing.These subsets were generated randomly from the available data.The system was said to be converged if the validation rootmean-square error (RMSE) did not improve over 300 iterations.
A hyperparameter screening was performed in order to determine the optimal network architecture, wherein it was chosen to use the same architecture for the atomic networks for each unique element.A penalty function, as introduced previously was used to determine the optimum.
Where ε T r the error of the training set, ε V the error of the validation set and ε T e the error of the test set.This allows identification of the architecture corresponding to the lowest overall error (characterised by the training set) while accounting for subset biasing, overfitting and underfitting by considering the offset against the validation and test sets.The hyperbolic tangent was used as the non-linear function.(Sigmoids were included in the optimisation, but were consistently outperformed.)The number of layers was allowed to range from 1 to 5. The number of nodes was varied from 10 to 50 using increments of 5.For each configuration, 50 networks were generated to account for variance during the fitting.ANN were used to perform the energy calculations for the GA.The GA was configured to generate 40 structures per cycle.The GA exploration was continued until at least 1000 clusters were generated.Once this target was achieved, the GA was set to converge when the LEME remained unchanged for 10 cycles.During the exploration, candidate clusters are analysed to identify extrapolation domains.If, during the GA cycles, structures were encountered that differ significantly from the dataset, the energy of these samples was verified using DFT.Structures were compared based on their CSR distance: G denotes the complete ANN input vector, a denotes the candidate cluster and b indexes the dataset samples.Permutational variance of atoms within the clusters was accounted for by concatenating like species in order of their vector norm.The CSR metric is used to identify the sample with the closest-matching fingerprint in the known dataset.Structures with a minimum distance exceeding 20% were verified using DFT.If the sample error exceeded the 2σ-range of the ANN, it was added to the dataset and the ANN was re-parametrised.The process continued until the structure exploration had converged.

Genetic Algorithm
A GA search was employed in order to obtain the set of structures near the energy minimum.To initialise the GA, initial candidate structures are obtained from the LEME of the transfer dataset.These clusters are adsorbed onto the selected support by utilising a geometric potential. 1The GA will subsequently perform modifications to the candidate clusters while searching for the global energy minimum.First, clusters in the active cycle are subject to geometry optimisation.This converges the cluster to a local minimum.Convergence of the GA is then evaluated.The structure of the candidates is compared to the GA population.Clusters are considered to belong to the same minimum if their energy differs by less than 0.1 eV and the CSR distance between them is less than 5%.Overlapping clusters are removed from the population.New candidate structures are generated by merging features from past candidates.Two parent clusters are selected by sampling an exponential distribution around the global minimum-energy candidate: The selected clusters are bisected by defining a dividing plane through each structure.This is done by randomly generating a surface normal through the geometric centre.One half from each parent is then used to generate a new candidate.Re-connecting these halves occurs along an inter-planar geometric potential to avoid high-strain configurations.Additional modifications to the candidate structures are performed to increase the sample diversity.4 operations have been defined.
1.The cluster is translated along the support surface.
2. The cluster is rotated relative to the support surface.
3. A vibrational distortion is applied to the structure following the approach detailed in earlier work. 1 4. Positions of the alloy constituents are shuffled.This option is disabled for mono-metallic clusters.
Each of these operations is assigned a 10% occurrence rate.The remaining 60% of candidates is left intact.

Network Performance
The optimised network architectures for each system are provided in Tables S1-S12.
Table S1: Optimised architectures for ANN of PtSn/Al 2 O 3 .Fitting was performed using a transfer dataset initialised on Pt 8 .RMSE and biases are reported in meV/atom.N denotes the number of sequential transforms in the network.W denotes the output dimension of the linear transforms.

LEME Structures
LEME trajectories for mono-metallic nanoclusters are given in Figures S1-S3.
Reference DFT-driven GA searches for supported PtSn nanoclusters are given in Figure S4.Geometries for the low-energy structures of the various nanocluster systems are given in Figures S5-S42.

DFT-ANN LEME Comparisons
In order to quantify the difference in cluster coordinates predicted by DFT and ANN, the Frobenius norm of the difference matrix is computed.
r denotes the coordinate vector of each atom in the cluster.Inter-cluster atomic ordering is determined through sequential minimisation of interatomic distances to account for indexing differences between samples.
Because the clusters in the LEME sets differ between DFT and ANN predictions, mapping of clusters is performed using CSR matching as detailed earlier.
The corresponding indices are reported alongside the norms.

Unsupported Co Clusters
A DFT-driven GA was performed to initialise a transfer dataset on unsupported Co 8 clusters.This dataset was extended to clusters of 6 to 20 atoms through ANN-driven GA initialised on geometries obtained from geometric adatom adsorption.Results are reported in Figure S43.Cost reduction of the transfer method compared to DFT-driven GA searches is similar to that of the SiO 2supported clusters.The cost of the DFT-driven GA for larger clusters increases more rapidly compared to the supported systems.This is attributed to the lack of a templating effect by the support, enhancing sample diversity.As a result, cost reduction from the ANN increases as particles grow larger.

FigureFigure S2 :
Figure S1: Low-energy structures obtained through ANN-driven GA searches for supported Co 8 .

Figure S3 :Figure
Figure S3: Low-energy structures obtained through ANN-driven GA searches for TiO 2 -supported clusters.

Figure S5 :
Figure S5: Geometries of the 10 lowest-energy clusters obtained through ANNdriven GA searches for Pt 8 /Al 2 O 3 .

Figure S13 :
Figure S13: Geometries of the 10 lowest-energy clusters obtained through ANNdriven GA searches for Sn 8 /Al 2 O 3 .

Figure S14 :
Figure S14: Geometries of the 10 lowest-energy clusters obtained through ANNdriven GA searches for Pt 8 /CeO 2 .

Figure S22 :
Figure S22: Geometries of the 10 lowest-energy clusters obtained through ANNdriven GA searches for Sn 8 /CeO 2 .

Figure S23 :
Figure S23: Geometries of the 10 lowest-energy clusters obtained through ANNdriven GA searches for Ni 8 /In 2 O 3 .

Figure S24 :
Figure S24: Geometries of the 10 lowest-energy clusters obtained through ANNdriven GA searches for Ni 7 In 1 /In 2 O 3 .

Figure S25 :
Figure S25: Geometries of the 10 lowest-energy clusters obtained through ANNdriven GA searches for Ni 6 In 2 /In 2 O 3 .

Figure S26 :
Figure S26: Geometries of the 10 lowest-energy clusters obtained through ANNdriven GA searches for Ni 5 In 3 /In 2 O 3 .

Figure S27 :
Figure S27: Geometries of the 10 lowest-energy clusters obtained through ANNdriven GA searches for Ni 4 In 4 /In 2 O 3 .

Figure S28 :
Figure S28: Geometries of the 10 lowest-energy clusters obtained through ANNdriven GA searches for Ni 3 In 5 /In 2 O 3 .

Figure S29 :
Figure S29: Geometries of the 10 lowest-energy clusters obtained through ANNdriven GA searches for Ni 2 In 6 /In 2 O 3 .

Figure S30 :
Figure S30: Geometries of the 10 lowest-energy clusters obtained through ANNdriven GA searches for Ni 1 In 7 /In 2 O 3 .

Figure S31 :
Figure S31: Geometries of the 10 lowest-energy clusters obtained through ANNdriven GA searches for In 8 /In 2 O 3 .

Figure S32 :
Figure S32: Geometries of the 10 lowest-energy clusters obtained through ANNdriven GA searches for Co 8 /Al 2 O 3 .

Figure S33 :
Figure S33: Geometries of the 10 lowest-energy clusters obtained through ANNdriven GA searches for Co 8 /CeO 2 .

Figure S34 :
Figure S34: Geometries of the 10 lowest-energy clusters obtained through ANNdriven GA searches for Co 8 /SiO 2 .

Figure S37 :
Figure S37: Geometries of the 10 lowest-energy clusters obtained through ANNdriven GA searches for Rh 8 /CeO 2 .

Figure S38 :
Figure S38: Geometries of the 10 lowest-energy clusters obtained through ANNdriven GA searches for Ru 8 /CeO 2 .

Figure S43 :
FigureS43: Relative cost of the transfer dataset method for GA searches of unsupported Co nanoclusters compared to DFT-driven GA.ANN were initialised using structures obtained through adatom adsorption along a geometric potential.Median values are reported.The cost of DFT post-optimisations is not included.

Table S4 :
Optimised architectures for ANN of PtSn/Al 2 O 3 .Fitting was performed using a transfer dataset initialised on PtSn/Al 2 O 3 and PtSn/CeO 2 .RMSE and biases are reported in meV/atom.N denotes the number of sequential transforms in the network.W denotes the output dimension of the linear transforms.

Table S5 :
Optimised architectures for ANN of PtSn/CeO 2 .Fitting was performed using a transfer dataset initialised on PtSn/Al 2 O 3 and PtSn/CeO 2 .RMSE and biases are reported in meV/atom.N denotes the number of sequential transforms in the network.W denotes the output dimension of the linear transforms.

Table S7 :
Optimised architectures for ANN of PtSn nanoclusters.Fitting was performed using a transfer dataset initialised on NiIn/In 2 O 3 .RMSE and biases are reported in meV/atom.N denotes the number of sequential transforms in the network.W denotes the output dimension of the linear transforms.

Table S8 :
Optimised architectures for ANN of NiIn/In 2 O 3 .Fitting was performed using a transfer dataset initialised on PtSn data.RMSE and biases are reported in meV/atom.N denotes the number of sequential transforms in the network.W denotes the output dimension of the linear transforms.

Table S9 :
Optimised architectures for ANN of PtSn nanoclusters.Fitting was performed using a transfer dataset initialised on PtSn and NiIn data.RMSE and biases are reported in meV/atom.N denotes the number of sequential transforms in the network.W denotes the output dimension of the linear transforms.

Table S11 :
Optimised architectures for ANN of Co 8 .Fitting was performed using a transfer dataset initialised on PtSn and NiIn data.RMSE and biases are reported in meV/atom.N denotes the number of sequential transforms in the network.W denotes the output dimension of the linear transforms.

Table S12 :
Optimised architectures for ANN of mono-metallic nanoclusters.Fitting was performed using a transfer dataset initialised on PtSn and NiIn data.RMSE and biases are reported in meV/atom.N denotes the number of sequential transforms in the network.W denotes the output dimension of the linear transforms.

Table S13 :
Comparison of LEME cluster geometries from GA searches using DFT and ANN.Coordinates are compared using the Frobenius norm of the difference matrix.Values are reported in Å. Inter-cluster atomic ordering is determined through sequential minimisation of interatomic distances.I denotes the index of the cluster in the LEME sets.Pt 8 Pt 6 Sn 2 Pt 4 Sn 4 Pt 2 Sn 6 Sn 8 I DFT I ANN ||F || I ANN ||F || I ANN ||F || I ANN ||F || I ANN ||F ||