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Stepwise Evolution of Photocatalytic Spinel-Structured (Co,Cr,Fe,Mn,Ni)_{3}O_{4} High Entropy Oxides from First-Principles Calculations to Machine Learning

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^{2}

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## Abstract

**:**

_{3}O

_{4}, for stoichiometric and non-stoichiometric structures. The effects of site occupation by different metal cations in the spinel structure are obtained through first-principles calculations and ML predictions. Our predicted results show that the lattice constants of these spinel-structured oxides are composition-dependent and that the formation energies of those oxides containing Cr atoms are low. The computing time and computing energy can be greatly economized through the tandem approach of first-principles calculations and ML.

## 1. Introduction

_{2}O

_{4}, where metal cations at A sites occupy the center of the tetrahedral position, metal cations at B sites occupy the center of the octahedral position, and anions locate at the vertexes of the polyhedron. According to the distribution of cations, spinel can be divided into normal spinel [27,28] and inverse spinel [28,29,30]. For the multi-component spinels, a complicated local structure is expected to persist. Spinel-structured HEOx were firstly synthesized and studied for many fantastic properties [14,23,24,31,32,33,34]. A single-phase (Fd3̅m) spinel-structured HEOx, (Co,Cr,Fe,Mn,Ni)

_{3}O

_{4}, was synthesized [14] in 2018. The defect structures and chemical diffusion in spinel-structured (Co,Cr,Fe,Mn,Ni)

_{3}O

_{4}were further proposed [24], the findings of which indicated the complexity of the defect structure in these high-entropy spinels.

## 2. Methods

#### 2.1. First-Principles Calculations

_{1}/amd) with a = 5.903 Å and c = 8.348 Å and FCO structures (Imma) with a = 6.002 Å, b = 6.017 Å, and c = 8.301 Å were set. A 520 eV plane wave cutoff and the 3 × 3 × 3, 4 × 4 × 3, and 4 × 4 × 3 Monkhorst-Pack k-point grid were used for the calculations. Structure cards for VASP were shown in the Supplementary Information.

_{2}O

_{4}), ${E}_{i}$ is the total energies of each element i in its stable phase, and ${C}_{i}$ is the concentration of each element.

#### 2.2. Machine Learning (ML) Algorithms

_{3}O

_{4}in A or B sites were set as the independent variables, X, and the lattice constants and formation energy of the spinels from DFT calculations were set as the dependent variables, Y, as an input file. Next, the hidden layers were used to capture the characteristics of the training data and reflect on the relevant parameters. The number of hidden layers was set to 2, and the number of neurons in each layer was set to 20 based on the decreasing mean square error in the trial and error process. Lastly, the output layer was used to represent the output variables of the network. High training and testing accuracy was achieved by modifying the training and parameters.

## 3. Results and Discussion

#### 3.1. Decision Flow

#### 3.2. ML-Model Selection and Performance

_{i}and ŷ

_{i}are the real and predicted values, respectively. The number of learning cycles is set to 1 million to ensure that the RMSE has a good convergence. The RMSE values of the lattice constants obtained from the BPN and GANN models decrease as the training progresses and finally converge to 0.0394 and 0.0071, respectively. The RMSE values of the formation energies obtained from the BPN and GANN models also decrease as the training progresses and converge to 0.0831 and 0.0182, respectively. The latter model, GANN, thus is better than the BPN model. The scatter plots of the lattice constants, a, b, and c obtained from the BPN and GANN models are shown in Supplementary Figure S3c,d, respectively. Supplementary Figure S3e,f are the scatter plots of the formation energies obtained from the BPN and GANN models, respectively. The data from the latter model is closer to the regression line than that of the former one, which means the GANN model has higher accuracy in this case. Therefore, we use the GANN model as the ML method in the following studies.

#### 3.3. Prediction of the New Spinel Structure

_{3}O

_{4}into two categories according to different metal cations in the A and B sites (Figure 2 and Figure 4), and the different proportion of metal cations in the A and B sites (Figure 3 and Figure 5) in the same composition. Figure 2 shows the predicted lattice constants, where the metal elements in the A and B sites of the spinel structure can be a single metal element or multiple metal elements. The colors represented the lattice constants a, b, and c are predicted from the GANN model. Lattice constants a, b, and c have different but similar values.

_{3}O

_{4}is set as 1 − x (y), as shown in Supplementary Table S3. The colors shown in Figure 3 represented the lattice constants a, b, and c are predicted from the GANN model. The results show that the lattice constants a, b, and c have different but similar values.

#### 3.4. Comparison of the Calculated and Predicted Results

_{2}O

_{4}, including cubic, tetragonal, and orthorhombic structures are taken into consideration, and they have 56, 28, and 28 atoms in a unit cell, respectively. The cubic, tetragonal, and orthorhombic structures of stoichiometry/non-stoichiometry AB

_{2}O

_{4}are also considered by installing a single cell and a supercell. The computing time required for DFT calculations differs from the degrees of difficulty in the self-consistent calculation of the electronic density matrix. The lattice constants and formation energies of stoichiometry/non-stoichiometry AB

_{2}O

_{4}predicted using ML are consistent with those calculated using DFT directly. ML also has the technological advance to deal with the crystallographic in a non-stoichiometric form. Thus, computing time and power can be saved, and the process is more efficient.

## 4. Conclusions

## Supplementary Materials

_{2}O

_{4})

^{42}. Table S3: The HEOx compositions of each condition. Table S4 HEOx model for prediction and calculation.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**A schematic of the main concepts of the experiment. The blue region indicates that the physical properties of the binary and ternary spinel compounds are calculated by using the DFT calculations. The green area indicates that the non-stoichiometric compound to be calculated requires the creation of a stoichiometric compound first; thus, the computational load is heavy. The red region indicates that the calculated physical properties of binary and ternary spinel compounds are used as machine learning input files, and then the predicted physical properties of the non-stoichiometric compound are obtained by using machine learning.

**Figure 2.**The predicted lattice constants of the spinel-structured HEOx with different metal cations in the A and B sites.

**Figure 3.**The predicted lattice constants of the spinel-structured HEOx with different proportions of metal cations in the A and B sites.

**Figure 4.**The predicted formation energies of the spinel-structured HEOx with different metal cations in the A and B sites.

**Figure 5.**The predicted formation energies of the spinel-structured HEOx with different proportions of metal cations in the A and B sites.

**Figure 6.**(

**a**) Lattice constants and (

**b**) formation energies from the DFT calculation (dark grey) and ML prediction (light grey).

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**MDPI and ACS Style**

Lin, C.-C.; Chang, C.-W.; Kaun, C.-C.; Su, Y.-H.
Stepwise Evolution of Photocatalytic Spinel-Structured (Co,Cr,Fe,Mn,Ni)_{3}O_{4} High Entropy Oxides from First-Principles Calculations to Machine Learning. *Crystals* **2021**, *11*, 1035.
https://doi.org/10.3390/cryst11091035

**AMA Style**

Lin C-C, Chang C-W, Kaun C-C, Su Y-H.
Stepwise Evolution of Photocatalytic Spinel-Structured (Co,Cr,Fe,Mn,Ni)_{3}O_{4} High Entropy Oxides from First-Principles Calculations to Machine Learning. *Crystals*. 2021; 11(9):1035.
https://doi.org/10.3390/cryst11091035

**Chicago/Turabian Style**

Lin, Chia-Chun, Chia-Wei Chang, Chao-Cheng Kaun, and Yen-Hsun Su.
2021. "Stepwise Evolution of Photocatalytic Spinel-Structured (Co,Cr,Fe,Mn,Ni)_{3}O_{4} High Entropy Oxides from First-Principles Calculations to Machine Learning" *Crystals* 11, no. 9: 1035.
https://doi.org/10.3390/cryst11091035