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Combining Generative Adversarial Network and First-Principles Based Microkinetics for Heterogeneous Catalyst Design

Microkinetic analysis based on density functional theory (DFT) was combined with a generative adversarial network (GAN) to enable artificial proposal of heterogeneous catalysts based on the DFTcalculated dataset. The approach was applied to the NH3 formation reaction on Ru-Rh alloy surfaces as an example. The NH3 formation turnover frequency (TOF) was calculated by DFT-based microkinetics. Specifically, six elementary reactions (N2 dissociation, H2 dissociation, NHx (x = 1–3) formation, and NH3 desorption) were explicitly considered, and their reaction energies were evaluated by DFT. On the basis of TOF values and atomic compositions, new alloy surfaces were generated using the GAN. This approach successfully generated the surfaces not included in the initial dataset but have higher TOF values. The N2 dissociation reaction was more exothermic for the generated surfaces, leading to higher TOF. The present study shows that automatic improvement of catalyst materials is possible by using the iterative steps of DFT calculation and sample generation by GAN.


Introduction
Catalysts play a crucial role in energy and environmental science, and its performance is often evaluated by the reaction rate or turnover frequency (TOF). Researchers have devoted great efforts to find new catalysts with higher reaction rates. It is well known that the reaction rate is governed by several factors: the activation energies, the number of active sites (or surface area of the catalyst), sticking coefficients on the surface, etc. Unfortunately, accurate measurement of such quantities requires both special expertise and great effort; therefore, detailed kinetic profiles have only been clarified for limited cases.
In the last few decades, theoretical simulation and computational methods became feasible alternatives to evaluate reaction kinetics. Ab initio or first-principles calculation is a popular one, because it provides atomic-scale information without requiring experimental data. For example, this approach can be helpful in identifying the active site on the catalyst surface, which is a fundamental issue in catalysis. Owing to recent developments in algorithms and computational resources, such atomic-scale simulations, especially those using density functional theory (DFT), are widely used nowadays.
While computational methods are useful for studying given real or proposed materials, it cannot automatically suggest new ones. Recently, it was realized that applying machine learning to computational chemistry could lead to computation-based material proposal. 1 Several promising examples have been reported in catalysis. [2][3][4][5] Among the several possibilities of combining computational chemistry and machine learning, the present author considers the so-called generative model a particularly important machine learning algorithm because it enables "extrapolative" proposal in the material or configuration space; that is, the search is not confined within given dataset. As a type of generative model, the generative adversarial network (GAN) is widely used, especially in artificial image generation. 6,7 Several groups have reported the application of GAN to material science; Kim et al. used it to find new zeolite systems, 8 and several groups also used it to artificially generate crystal 3 structures with desirable properties. 9,10 A catalytic reaction often involves several species and elementary reactions. Microkinetic analysis explicitly treats a set of elementary reactions. Therefore, it is often more accurate than the kinetic analysis based on the global rate expression. 11 Currently, DFT-based microkinetics is widely used in catalysis research because it is a powerful tool since kinetic information, such as reaction energy and activation barrier, can be calculated by DFT. [11][12][13][14][15] Considering this, the combination of DFT calculations, microkinetics, and catalytic material generation from generative models is a promising approach toward the rational design of catalysts.
The present paper describes a new approach based on DFT calculations and sample generation by GAN for heterogeneous catalyst search. The generation procedure is extrapolative, because the proposed catalytic material need not be included in the initially prepared dataset. Here, the GAN part aims to generate materials with a high TOF for the target catalytic reaction, where the TOF is calculated using DFT-based microkinetics. The ammonia (NH3) synthesis reaction, known as the Haber-Bosch process, is considered in this study as a representative heterogeneous catalytic reaction. [16][17][18] Below, details of the DFT-GAN procedure are described in the Methods section, and its performance is discussed in the Results and Discussion section.

a. Models and details of DFT calculation
Here, catalytic NH3 synthesis is assumed to occur on a Rh-Ru bimetallic alloy surface. The Ru stepped surface was constructed first, and the bimetallic alloys were constructed by replacing Ru atoms with Rh atoms. Stepped metal surfaces were considered because NH3 formation is known to occur there. 17,19 The positions of the Ru atoms replaced by Rh atoms were chosen randomly in the original dataset, while the position of the Rh atoms is determined by the GAN; the details will be discussed later. The original dataset included 100 metal surfaces. The metal surfaces are modeled by repeated 4 slabs, and the stepped surface is modeled by removing half of the atoms in the top layer. Each slab consists of a (6×4) supercell in the lateral direction, with four atomic layers in the z-direction.
Consequently, 84 atoms are included in the model. The typical structure of the surface model is shown in Fig. 1. The adsorption positions of N, H, NH, and NH2 were assumed to be the fcc three-fold hollow site and atop adsorption was assumed for NH2 and NH3, as these positions are the most stable adsorption sites on the Ru-stepped surface. 20 These adsorption sites are shown in Fig.1.
The BEEF-vdW exchange-correlation functional was used in the DFT calculations because it provides an accurate description of the van der Waals interaction. 21 The core electrons were represented by the projector-augmented wave (PAW) potentials. 22 The valence electrons were expanded with plane waves up to a cutoff energy of 400 eV. Spin polarization was included throughout, and no symmetry constraint was imposed on the geometries. A Gaussian scheme was used in the smearing of electron occupation near the Fermi level. The convergence thresholds for the electronic and geometry Optimization of the unit cell lengths was performed for all surfaces in the absence of the adsorbate molecules. Brillouin zone integration was performed by numerical integration at the gamma point. A vacuum layer of ~12 Å was placed between the slabs, and dipole correction was applied in the zdirection to cancel the artificial interaction between slabs. All DFT calculations were performed using the Vienna ab initio simulation package (VASP) version 5.4.4. 23, 24

b. Elementary reactions and rate of NH3 formation
The overall reaction of NH3 synthesis (1) is generally considered to include the following six elementary reactions. 25 (2) where the asterisk (*) denotes a vacant active site on the metal surface, and the species with asterisks are the adsorbed species. The reaction energies in eq. (2)-(7) were determined from the total energy calculated by DFT, that is, the sum of electronic and nuclear repulsion energies.
Previous experimental and theoretical work suggests that the rate-determining step (RDS) is eq.
(2), namely the dissociative adsorption of N2. 17,26 The present study followed this assumption, and the other reaction steps were equilibrated. In this case, the fractional surface coverage of adsorbed species where pi is the partial pressure of NH3 or H2, and Ki is the equilibrium constant of eqs.
Ea is the activation energy of eq. (2), A is the pre-exponential factor with the value of 0.
The values of = 0.87 and = 1.34 for the stepped metal surface were taken from the literature. 28 The calculation was carried out at T = 700 K and a total pressure of 100 bar. N2 and H2 in the inlet gas were stoichiometric, i.e., pN2 : pH2 = 1 : 3. The conversion of N2 was set to 10%.

c. Details of the GAN
Similar to the original and several extended versions of GAN, the entire system here consists of the generator (G) and discriminator (D) networks. In the present case, fake samples with a high NH3 formation rate are desirable, and they can be generated using G. Therefore, the conditional GAN (CGAN) was applied because it enables the generation of fake samples corresponding to a given label. (1) DFT calculation is performed to obtain Ea and ΔE values for elementary reactions of eq. (2)-(7). This is done for all samples in the dataset.
(2) The TOF for NH3 formation is calculated according to eq. (9) using the DFT-calculated ΔE and Ea values.
(3) The calculated samples are sorted according to the DFT values and grouped into several classes (n) according to the NH3 formation rate. Here, the number of groups is set to five, and the group with the highest TOF is labeled n = 1.
(4) Networks D and G are trained with the dataset using the backpropagation scheme.
(5) G generates fake samples for n = 1. Any generated surface that overlaps with the existing sample set is removed.
(6) DFT calculation is performed for the newly generated samples. The results are added to the dataset for use in the next iteration.
Note that the size of the dataset increases as the iterations proceed. This feature is favorable in terms of training neural networks, as a larger number of samples can be used in the training. Several studies have shown that such iterative training of the GAN is effective. 30,31 When training D and G, the loss function was set to the mean-squared error in both cases, and the ADAM optimizer was used. The learning rate was set to 1.0 × 10 −3 , and the parameters 1 and 2 were set to 0.5 and 0.999, respectively. The dropout rate was set to 0.3. The minibatch size was set to 20% of the sample size at each iteration. The maximum number of the training process (i.e., the epoch) was set to 2000.
The Python library atomic simulation environment (ASE) was used to construct the model and perform the DFT calculations. 32 The GAN part was calculated with PyTorch version 1.8. These Python codes are freely available on the author's GitHub page. 33

Results and Discussion
First, 100 bimetallic alloy surfaces were generated by randomly replacing Rh atoms with Ru atoms on a surface. DFT calculations were performed on these samples to obtain the TOF values. In the following, this dataset is called the "original dataset." Then, the iterative GAN procedure described above was applied. Five iterations were performed, meaning that four generations of new samples were created in addition to the original dataset. this TOF value is more than ten times larger than the best value in the original dataset. These facts show that the GAN successfully generated a metal surface with high catalytic performance in an extrapolative manner. The TOF values at each iteration are summarized in the so-called violin plot in Figure 3(b). The violin-shaped curves show the probability density of the TOF values, and the box inside each curve indicates the quartiles. The plot shows that the original dataset has widely distributed TOF values. The TOF value distribution of the generated surfaces (iter = 1-5) are more skewed toward the high TOF region. This trend clearly shows that the GAN is much more efficient than the random sampling, to obtain the metal surface with high NH3 formation rate.

10
The property of the generated surface agrees with the existing chemical or physical insights. It is widely known that NH3 formation is much faster on the Ru surface than on the Rh surface. 34 In this study, surfaces generated by GAN tend to have a high proportion of Ru; the full dataset can be found in Supporting Information. This shows that the neural network captures the experimental tendency of transition metal elements by learning from the TOF of NH3 formation calculated by DFT. The generator loss (G-loss) and discriminator loss (D-loss) during the training process are plotted in Figure 4. Since each iteration has 2000 epochs, there are 10000 epochs in five iterations. At the earlier stage of training, G-loss is smaller than D-loss. However, after ~1000 epochs, D-loss becomes lower than G-loss, meaning that the D part of the GAN is well-trained.
To understand why the generated surfaces have higher TOFs, the energetics of NH3 formation reaction were analyzed. Figure 5 summarizes the potential energy curves, reaction energies of the elementary reactions, and surface fractional coverages of the adsorbates. Here, the Rh-Ru surfaces with the best TOF values at iter = 0, 3, 4, and 5 are compared. The compositions of these surfaces are Rh8Ru76, Rh4Ru80, Rh12Ru72, and Rh8Ru76, respectively. The best TOF value (1.1 × 10 −3 site −1 ·s −1 ) is that of Rh8Ru76, which is generated at iter = 5. This value is much higher than that of Rh8Ru76 at iter = 0 (1.0 × 10 −4 site −1 ·s −1 ), which is the highest TOF in the original dataset (for brevity, these surfaces are denoted as Rh8Ru76-iter5 and Rh8Ru76-iter0 in the following discussion). Note that these two surfaces have the same composition, but the position of the Rh atoms are different each other.  show a deep potential energy sink at the NH2* + N* + 4H* state caused by the exothermic formation of NH and NH2 and endothermic formation of NH3. This energy sink is unfavorable from the thermodynamic viewpoint because the high stability of NH* and NH2* leads to surface poisoning. The lack of vacancy in the active site prohibits the next catalytic reaction; this issue will be discussed later.
The potential energy curves for iter = 2-4 have lower potential energy at NH2* + N* + 4H*, similar to that for iter = 0. However, at iter = 5, the endothermicity of the reaction is much improved. This is beneficial in terms of the accessibility of the N2 molecule on the active site. Thus, the potential energy curves show that the GAN-generated surfaces successively improve the thermodynamic character of NH3 formation.
Consequently, the activation barrier Ea becomes lower in the latter owing to the linear free energy relationship (eq. (11)). Another notable feature is the ΔE for NH3 formation. As stated above, the facile formation of NH3 alleviates surface poisoning by NH2. The Figure 5(b) shows that ΔE for NH3 formation progressively becomes less endothermic as the iteration proceeds. This suggests that the GAN improves not only the kinetics but also the thermodynamics of the NH3 formation. and the vacant site (i.e., the active site) over several Rh-Ru surfaces. A notable difference is seen in the coverage of vacant site (θvac); for example, θvac is 7.5 × 10 −3 for Rh8Ru76-iter5, but 3.9 × 10 −3 for Rh8Ru76-iter0. The higher θvac of Rh8Ru76-iter5 facilitates NH3 formation by leaving accessible active sites for the N2 dissociation reaction. In accordance with its higher θvac, Rh8Ru76-iter5 has a lower θNH2 (0.73) than Rh8Ru76-iter0 (0.89). NH2 is the most abundant adsorbate during NH3 formation. A high 13 θNH2 indicates NH2 poisoning on the surface to reduce the NH3 formation rate, which is known to be a serious disadvantage of Ru surfaces. 35 Thus, lowering the θNH2 and θH values is desirable for the catalytic performance. The present data show that the GAN-generated surface has a lower θNH2 value than those included in the original dataset.
All these results show that the proposed DFT-GAN improves the TOF for NH3 formation by adjusting the detailed energetics of the elementary reactions. Even though the GAN was not explicitly provided with such information, training the neural networks with DFT data successfully captured these details.

Conclusions
The present paper proposes a new approach combining computational chemistry and machine learning to generate new catalytic surfaces in an extrapolative manner. Density functional theory (DFT) is used to calculate the energies of elementary reactions on a provided set of catalytic materials.
The results are fed into a generative adversarial network (GAN) to propose additional materials. Here, the approach was used to enhance the turnover frequency (TOF) of NH3 synthesis in the Rh-Ru bimetallic alloy surface system. The DFT-GAN iterations consist of the following steps. The iterative process was started with 100 stepped alloy surfaces generated by random atomic replacement. After five iterations, Rh8Ru76 was successfully obtained as a surface not in the original dataset. The TOF of the generated surface was more than ten times higher than the best TOF value in the original dataset. Overall, samples generated in later iterations tend to have higher TOF, indicating that the iterative DFT-GAN scheme helps train the neutral networks in GAN. Also, the generated surfaces generally have a higher proportion of Ru atoms, which agrees with the experimental fact that the Ru surface is a far better catalyst than the Rh surface. The generated surfaces have higher TOF because of (a) a lower N2 dissociation reaction energy (which reduces activation energy for the ratedetermining step) and (b) a lower energy in NH3 formation (which reduces NH2 coverage on the surface and alleviates NH2 poisoning). The present study shows that the combination of DFT and GAN 16 is a promising strategy for the automatic and continuous improvement of catalyst performance.