English

• With the development of machine learning, using theoretical calculations to systematically predict new material structures has received widespread attention. Crystal structure prediction (CSP), as an effective material design method, can quickly find potential crystals in theory [1]. The traditional method combines density functional theory (DFT) and structure search algorithms, iteratively optimizes atomic positions along the energy gradient, and determines stable structures corresponding to global or local minimum energy [2,3]. The idea is of great importance in the field of new material design, but the complex and time-consuming process of energy calculation using DFT has largely limited the development of crystal structure prediction. But fortunately, machine learning can be a good solution to such problems [4].

  Machine learning uses DFT attribute databases (such as Materials project, AFLOW, OQMD, and MatB) to build relationships between crystal structures and properties, which can greatly reduce computational costs. At present, the models for predicting material properties mainly include: SchNet [5], crystal graph convolutional neural network (CGCNN) [6], improved crystal graph convolutional neural network (iCGCNN) [7], atomic line graph neural network (ALIGNN) [8], Material Graph Neural Network (MEGNet) [9], Geometric Information Enhanced Crystal Graph Neural Network (GeoCGNN) [10]. Through these machine learning models, the physical properties of crystalline materials can be predicted, such as formation energy (∆H) [11,12], Gibbs free energy [13], band gap [14], Atomization energy [15,16], electron density [17], thermal conductivity [18,19]. Most material property prediction models are based on graph neural networks, which represent molecular or crystalline materials as graphs, the most common of which are using element properties as node features, and interatomic distances or bond valences as edge features. For example, Cheng et al. [20] proposed a crystal structure prediction framework with graph networks and optimization algorithms, which utilizes graph network models to predict the formation energy of crystals as an alternative to DFT calculations. By experimental comparison, the computational cost of the framework is reduced by 3 orders of magnitude compared to traditional prediction methods.

  In the search for new materials in uncharted chemical territory, it is not enough to consider the use of machine learning to improve the speed of new material prediction; it is important to improve the stability and synthesizability of new materials. Material stability studies play a crucial role in ensuring material properties. Currently, many crystal prediction software use optimization algorithms such as simulated annealing [21], genetic algorithm [22], particle swarm optimization [23], Bayesian optimization [24] and hybrid algorithm [25] in order to search for the lowest-energy structure in a crystal system. For example, the widely used crystal structure prediction software, CALYPSO, employs a particle swarm optimization algorithm where the optimization objective function is usually the total energy of the crystal [24,26]. CALYPSO searches for the crystal structure with the lowest energy by optimizing the lattice parameters and atomic positions to determine the most stable state. In the field of crystal structure prediction, it is often assumed that the lower the energy, the more stable the crystal structure is, but it should be noted that relying on energy alone to determine the stability of a crystal structure may overlook other factors.

  Considering multiple factors that affect the stability of the crystal structure in a comprehensive manner can help to explore new materials more fully and accurately. A more comprehensive assessment of the crystal structure is not limited to considering not only the total energy, but also multiple structural features such as energy band structure, electron density, and magnetic moment. For example, The Gator takes the total energy, band gap, and electron density of the crystal structure as the objective function, and uses the genetic optimization algorithm to search for the crystal structure [27,28]. Reference [29] proposed a crystal structure prediction method based on atomic contact map, by maximizing the matching between the contact map of the predicted structure and the real crystal structure, using global optimization algorithms to search Wyckoff positions. Reference [30] realized multi-objective optimization by optimizing the accuracy of contact map matching, individual age and coordination number matching, thus reconstructing crystal structures with higher quality and alleviating the problem of getting stuck in local optima. Although, there have been a number of researchers optimizing crystal structures from multiple physical factors, most of them consider the thermodynamic stability aspect and neglect the dynamics stability.

  To enable rapid energy evaluation and improve stability prediction, this paper proposes a crystal structure prediction approach that takes into account both thermodynamic and dynamic stability. Specifically, we use a pre-trained MEGNet model based on a materials database to predict the formation energy of the crystals to assess the thermodynamic stability, and compute the Lennard-Jones potential using an empirical potential function to assess the dynamics stability. On this basis, an objective function containing both the formation energy and the Lennard-Jones potential is constructed for structure optimization. In addition, we analyze the atomic bonding of the crystals using interatomic contact map to screen out the structures that are stable in interatomic bonding. Finally, the validity of the method proposed in this paper is verified by several sets of crystal structures and the stability of the predicted crystals is analyzed by phase diagrams and phonon spectra.

  The crystal structure prediction method proposed in this article can quickly predict a stable crystal structure based on the given chemical composition of the material. Its working principle is shown in Fig. 1.

  Figure 1

  

  Figure 1.  Workflow of stable crystal structure prediction based on deep learning and multi-objective optimization.

  DownLoad: Full-Size Img PowerPoint

  The prediction framework proposed in this paper consists of six modules: Importing chemical compositions, cell parameter representation, generating crystal structures, contact map screening, constructing objective functions and crystal structure optimization. First, the atomic species and number are imported and the crystal cell parameters are set in order to randomly generate the initial crystal structure. Based on this, contact map are obtained by calculating the atomic distances to screen the structures for bonding between atoms. Then, the formation energy and the Lennard-Jones potential are obtained using a pre-trained model and an empirical function, respectively, to construct the objective function. Finally, a Bayesian optimization algorithm is used to iteratively search for stable crystals with low formation energies and potentials and with bonded atoms.

  Our crystal structure prediction framework requires only the chemical composition of the crystal structure in the input file, such as Ca₄F₈ or K₄Cl₄. Running the structure search program completes the 5000-step optimization process in approximately 20 min, yielding relatively good results.

  What needs to be introduced in detail is the objective function, which is constructed by measuring thermodynamic stability parameters ∆H and dynamic mechanical stability parameters V. Among them, the ∆H can be obtained through the pre-trained graphical model, and the V is obtained through the empirical function. In the constructed objective function, the smaller ∆H is, the more stable the thermodynamics of the crystal is, and V needs to be close to 0. Its square can not only make V approach 0 infinitely, but also effectively eliminate the influence of singular points.

  In general, crystal structure characterization methods can basically be classified into three categories, which are local descriptors, global descriptors and topological descriptors. The descriptor of the crystal structure is the core of the prediction of the crystal structure. The current popular methods include the atomic center symmetric function descriptor (ACSF) [31], the atomic position smooth overlap descriptor (SOAP) [32], Voronoi Tessellation [33,34], Coulomb matrix [35,36] and so on. While these physics inspired descriptors can be used to effectively predict performance, the processing of these descriptors is complex. With the construction of large-scale material databases and the development of deep learning, graph network-based methods can represent crystal structures well, and graph nodes and edges can intuitively reflect the properties of each atom in the crystal. The crystal structure can be represented by vectors ν_i and e_k respectively for atomic properties and atomic bond properties, where iϵ(1,…,N), kϵ(1,…,M), N is the total number of atoms, and M is the total number of atom pairs.

  Similarly, the lattice parameters of the crystal can also be expressed by the vector L(a, b, c, α, β, γ), where the a, b, c are the side lengths of the unit cell along three different directions, and the α, β, γ are the angles between the unit cells. The macroscopic properties of the crystal structure (such as temperature, pressure) can also be represented by the vector u. In addition, the atomic bond e_k can be calculated from the distance d(i, j) between atoms:

  ek=ei,j=exp(−γ(di,j−μ))

  (1)

  where di,j=∥vi−vj∥, γ and μ are hyperparameters, which are Gaussian centers.

  The nodes of the crystal diagram correspond to the atomic properties ν_i of the crystal, the connecting node edges correspond to the atomic bond properties e_i of the crystal, and the global properties of the crystal diagram correspond to the macroscopic properties u of the crystal. Therefore, the crystal structure can be represented by the crystal diagram, which can be used as the input of the property prediction model.

  Crystal contact map can effectively demonstrate the contact between atoms or ions, and bonding between atoms or ions is one of the key contact modes [37]. Specifically, when two atoms are connected to each other through a covalent or ionic bond, an edge will appear in the contact map to represent the connection between them. This shows that the edges of the contact map not only reflect bonding relationships, but may also include non-bonding adjacent relationships, such as van der Waals forces. Therefore, we can analyze key information such as bonding patterns, bond lengths and bond angles between atoms through crystal contact map.

  The crystal structure is represented by parameters such as lattice parameters, atomic coordinate positions, and symmetry, which can be obtained through experimental measurements or simulation calculations. In a crystal structure, the distances between atoms are calculated to determine which atoms are in contact. Two atoms are considered to be in contact if the distance between them is less than some predefined threshold. For atomic pairs identified as being in contact, contact information can be visualized as a contact map, which can be used to further study structure, properties, and interactions in crystals. As shown in Fig. 2, there are 12 atoms in BPO4 in a single unit cell. Through the Wyckoff positions of B, P, and O atoms in BPO₄ and the corresponding x, y, and z coordinates, the 12 × 12 contact matrix corresponding to BPO₄ can be calculated.

  Figure 2

  

  Figure 2.  Representation of the crystal BPO4. (a) Crystal parameter table, (b) crystal structure diagram, (c) crystal contact map.

  DownLoad: Full-Size Img PowerPoint

  From a thermodynamic point of view, the lower the formation energy of a crystal, the less likely it is to decompose or transform into other phases, so the more stable the crystal is [38,39]. This article uses a graph neural network model to predict the formation energy of crystals to reduce the time cost of energy calculations. Specifically, the MEGNet model is trained and verified by using the stable and metastable crystal structure data in the 60,000 Materials Project, excluding benchmark crystals that can be used as predictable effect verification. In order to verify the accuracy of the predictive model, the 48,000 data are used for model training, and the remaining data are used for testing and verification.

  Crystal dynamic stability involves the dynamic behavior and phase transition characteristics of crystals, which can be evaluated through crystal surface energy, barrier height, crystal dynamics simulation and other methods. Among them, the Lennard-Jones potential function is a potential function commonly used in classical molecular dynamics simulations to describe intermolecular interactions. It contains attraction terms and repulsion terms and can simulate interactions between non-polarized molecules [40]. The mathematical expression of the Lennard-Jones potential function is as follows:

  V=4ε∑i<j[(δrij)12−(δrij)6]

  (2)

  where V is the potential energy between the two molecules, ε represents the strength of the attraction term, δ represents the effective diameter between the molecules, and r is the distance between the two molecules. The first term (δ/r_ij)¹² in this potential function represents the long-range attraction, which gradually decreases as the distance r between the two molecules increases. The second term (δ/r_ij)⁶ represents the short-range repulsive force, which increases rapidly as the distance r decreases.

  In summary, we construct two objective function that can minimize the formation energy and ensure that the Lennard-Jones potential approaches 0 to perform Bayesian optimization and search for a stable crystal structure. As shown in (3), (4), respectively:

  f=ΔH+|V|

  (3)

  where

  |V|={V,rij/σ<1−V,rij/σ<1

  f=ΔH+V2

  (4)

  where f is objective optimization function, ΔH is the formation energy, and V is the potential energy between molecules.

  In order to facilitate the optimization of the objective function f, we change the negative V value into a positive value through absolute value and square respectively. Compare the two methods |V| and V², it is exciting to note that there exists a case of V−′(1)≠V+′(1) for |V| when rij/δ→1, which shows that |V| is not conductible at that point. Whereas V² solves this problem nicely, that is, when rij/δ→1, V² is a smooth curve, which can show that V2 is differentiable at this point. More laudably, V² tends to 0 when rij/δ>1.

  The crystal contact map records the connection relationships between all atoms in the unit cell and can capture the interactions between atoms very well. Generally speaking, when the average number of contacts of atoms in a crystal is greater, the more stable the crystal structure is, especially ionic crystals and metal crystals. Taking NaCl as an example, Na⁺ and Cl⁻ combine to form crystals by electrostatic attraction, and their different crystal structures correspond to different contact patterns (Fig. 3).

  Figure 3

  

  Figure 3.  NaCl crystal contact map analysis. (a) Stable NaCl crystals and their corresponding crystal contact map in the material Project database, (b-d) NaCl crystals and their corresponding crystal contact map predicted by the method in this article.

  DownLoad: Full-Size Img PowerPoint

  Fig. 3a illustrates a stable NaCl crystal with the Material Project identifier mp-22,862, the contact map of the NaCl crystal predicted in this paper is consistent with it (Fig. 3d). Where the number of contacts for Na⁺ and Cl⁻ ions is both 24, and the contact map is symmetrical along the positive diagonal. In contrast, ions in Figs. 3b and c have contact numbers of 12 and 14, respectively, with contact map displaying symmetry only along the anti-diagonal. Additionally, the crystals corresponding to Figs. 3b and c have larger lattice parameters, indicating a greater crystal volume. As the crystal volume increases, the distances between atoms or ions change, thereby influencing the stability of the crystal. Therefore, it is meaningful to screen crystal structures by calculating the crystal contact map.

  To validate the effectiveness of the objective functions we have set, we selected the optimal structures obtained under different objective functions and performed structure optimization and property calculations using VASP for these three structures. Firstly, we analyzed the variations in crystal parameters after structure optimization for these three crystals. Additionally, we calculated the free energy and formation energy of these three crystal structures (Table 1). Finally, we analyzed the thermodynamic and dynamic stability of the crystal structures using the convex hull diagram and phonon spectrum (Fig. 4).

  Table 1

  

  Table 1.  Properties of optimal NaCl searched by three objective functions.

  DownLoad: CSV

  Compounds	Objective function	Lattice parameters	Lattice parameters (structure optimization)	Free energy TOTEN (eV)	Formation energy (eV/atom)
  Na4Cl4	f=ΔH	a = 6.45933 Å α = 90.0000°	a = 6.14466 Å α = 90.0000°	27.79312767	-1.88861
  		b = 6.45933 Å β = 90.0000°	b = 6.14466 Å β = 90.0000°
  		c = 6.45933 Å γ = 90.0000°	c = 6.14466 Å γ = 90.0000°
  	f=ΔH+|V|	a = 6.45317 Å α = 90.0000°	a = 6.14375 Å α = 90.0000°	27.79299033	-1.888593
  		b = 6.45317 Å β = 90.0000°	b = 6.14375 Å β = 90.0000°
  		c = 6.45317 Å γ = 90.0000°	c = 6.14375 Å γ = 90.0000°
  	f=ΔH+V2	a = 5.67849 Å α = 90.0000°	a = 5.67849 Å α = 90.0000°	28.72775854	-2.005439
  		b = 5.67849 Å β = 90.0000°	b = 5.67849 Å β = 90.0000°
  		c = 5.67849 Å γ = 90.0000°	c = 5.67849 Å γ = 90.0000°

  Figure 4

  

  Figure 4.  The stability analysis of NaCl crystal structures. (a) The optimal NaCl crystal structures obtained through three different objective functions. The convex hulls of NaCl crystals by (b) f = ΔH and (d) f = ΔH + V². The phonon spectra of NaCl crystals by (c) f = ΔH and (e) f = ΔH + V².

  DownLoad: Full-Size Img PowerPoint

  By comparing the data in Table 1, it is evident that the NaCl crystal obtained through the f = ΔH + V² objective function still maintains unchanged lattice parameters after VASP structure optimization, with the highest free energy and smaller formation energy. The stability of the crystal structure can be analyzed from two levels: molecular thermodynamics and molecular dynamics. The formation energy reflects the minimum energy required to form a certain compound, so using the formation energy criterion to evaluate the relative stability of a material is closer to the actual situation. It is easier to intuitively judge the thermodynamic stability or metastability of a crystal by using the convex hull diagram. It is generally believed that when the convex hull energy of a new crystal is less than 0.2 eV per atom, the structure can be judged to be stable. Figs. 4b and d show the convex hull diagrams of NaCl crystals predicted by the optimization functions f = ΔH and f = ΔH + V², respectively. It is evident that Fig. 4d features a red label "NaCl (No ID)", which indicates that the predicted crystal structure searched by f = ΔH + V² has an energy formation below 0.2 eV per atom, suggesting its stability. Additionally, there is no corresponding ID number in the Materials Project database, indicating that this is a newly discovered NaCl structure.

  On the other hand, the phonon spectrum of a crystal provides a means to evaluate the dynamic stability. If there is an imaginary frequency in the phonon spectrum, that is, when the frequency is a negative value, it means that the crystal structure is unstable in the corresponding vibration direction. As shown in Fig. 4e, there is no imaginary frequency in the phonon spectrum of the NaCl structure searched by f = ΔH + V², indicating that the crystal structure is stable. The phonon spectrum frequency of the NaCl structure searched by f = ΔH is less than 0 at G, which does not meet the molecular dynamics stability.

  In addition to the aforementioned experiments, this paper also validates the prediction results for compounds of A-type, AB-type, and ABC-type under three different objective functions: f = ΔH, ΔH+|V|, and f = ΔH + V², such as C, Si, NaCl, KCl, ZnO, TiO₂, HgCl₂, and K₂CO₃. According to the experimental results, it can be observed that the proposed method not only predicts a larger number of crystals but also yields crystal structures with smaller formation energies. This indicates that our method can indeed predict more stable crystal structures. For detailed information, please refer to the supplementary material section.

  In summary, we build a framework that effectively predicts stable crystal structures. The framework mainly includes parameter calculations that affect crystal stability, multi-objective optimization of crystal structures, and screening of atomic bonds within the crystal. In the field of crystal structure prediction, the stability of the crystal structure is the primary condition for experimental synthesis of crystal materials, so this article focuses on predicting stable crystal structures. The graph neural network model is used to replace the traditional DFT energy calculation part to reduce the time cost in the crystal structure prediction process. Through the comparison of energy prediction results, it can be seen that graph neural network model has the advantage of having a prediction accuracy close to DFT calculation and being less time-consuming. Also, this paper uses the formation energy and Lennard-Jones potential function as indicators of the thermodynamic stability and dynamic stability of the crystal structure respectively to construct the objective function. The contact map of the crystal is then used to screen out crystals where the atoms are bonded, which make the searched crystal structure more stable. According to the experimental results, the crystal structure prediction method we proposed can not only solve the problem of time-consuming DFT calculations, but also ensure the stability of the crystal to a certain extent.

  Declaration of competing interest

  The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

  CRediT authorship contribution statement

  Lu Li: Methodology, Formal analysis, Data curation. Jianing Shen: Software, Methodology. Qinkun Xiao: Writing – review & editing, Funding acquisition, Formal analysis. Chaozheng He: Writing – review & editing, Supervision, Resources, Funding acquisition, Formal analysis. Jinzhou Zheng: Supervision, Investigation. Chaoqin Chu: Validation, Supervision, Formal analysis. Chen Chen: Visualization, Formal analysis, Data curation.

  Acknowledgment

  This work was supported by the Nature Science Foundation of China (Nos. 61671362 and 62071366).

  Supplementary materials

  Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.cclet.2024.110421.

  
  
• 1. [1]

     X. Yuan, L. Li, Z. Shi, et al., Adv. Powder Mater. 1 (2022) 100026.

  2. [2]

     H. Jing, P. Zhu, X. Zheng, et al., Adv. Powder Mater. 1 (2022) 100013.

  3. [3]

     J. Gao, J. Zhong, G. Liu, et al., Adv. Powder Mater. 1 (2022) 100005.

  4. [4]

     Z. Yuan, T. Tian, F. Hao, et al., J. Power Sources. 609 (2024) 234697.

  5. [5]

     K.T. Schütt, H.E. Sauceda, P.J. Kindermans, et al., J. Chem. Phys. 148 (2018) 241722.

  6. [6]

     T. Xie, J.C. Grossman, Phys. Rev. Lett. 120 (2018) 145301.

  7. [7]

     C.W. Park, C. Wolverton, Phys. Rev. Mater. 4 (2020) 063801.

  8. [8]

     K. Choudhary, B. DeCost, Npj Comput. Mater 7 (2021) 185.

  9. [9]

     C. Chen, W. Ye, Y. Zuo, et al., Chem. Mater. 31 (2019) 3564–3572. doi: 10.1021/acs.chemmater.9b01294

  10. [10]

      J. Cheng, C. Zhang, L. Dong, et al., Commun. Mater. 2 (2021) 92. doi: 10.4103/rid.rid_22_21

  11. [11]

      Y. Mao, H. Yang, Y. Sheng, et al., ACS Omega 6 (2021) 14533–14541. doi: 10.1021/acsomega.1c01517

  12. [12]

      Y. Hinuma, S. Mine, T. Toyao, et al., ACS Omega 7 (2022) 18427–18433. doi: 10.1021/acsomega.2c00702

  13. [13]

      X. Hao, J. Liu, H. Luo, et al., Crystals 9 (2019) 256. doi: 10.3390/cryst9050256

  14. [14]

      R. Wang, Y. Zhong, X. Dong, et al., Inorg. Chem. 62 (2023) 4716–4726. doi: 10.1021/acs.inorgchem.3c00233

  15. [15]

      A. Khan, H. Tayara, K.T. Chong, et al., Comput. Mater. Sci. 220 (2023) 112063.

  16. [16]

      C. Chu, Q. Xiao, C. He, et al., Chin. Chem. Lett. 35 (2024) 109186.

  17. [17]

      Y. Sun, W. Hu, SmartMat 3 (2022) 474–481. doi: 10.1002/smm2.1074

  18. [18]

      R. Meenashi, K. Selvaraju, A.D. Stephen, et al., J. Mol. Struct. 1213 (2020) 128139.

  19. [19]

      C.M. Collins, L.M. Daniels, Q. Gibson, et al., Angew. Chem., Int. Ed. 60 (2021) 16457–16465. doi: 10.1002/anie.202102073

  20. [20]

      G. Cheng, X.G. Gong, W.J. Yin, et al., Nat. Commun. 13 (2022) 1492.

  21. [21]

      Z.L. Liu, Comput. Phys. Commun. 185 (2014) 1893–1900.

  22. [22]

      P. Avery, C. Toher, S. Curtarolo, et al., Comput. Phys. Commun. 237 (2019) 274–275.

  23. [23]

      Y. Wang, J. Lv, L. Zhu, et al., Comput. Phys. Commun. 183 (2012) 2063–2070.

  24. [24]

      M.S. Priyadarshini, O. Romiluyi, Y. Wang, et al., Mater. Horiz. 11 (2024) 781–791. doi: 10.1039/d3mh01474f

  25. [25]

      T. Yamashita, H. Kino, K. Tsuda, et al., Sci. Technol. Adv. Mater.: Methods. 2 (2022) 67–74. doi: 10.1080/27660400.2022.2055987

  26. [26]

      C. Tang, G. Kour, A. Du, Chinese Phys. B 28 (2019) 107306. doi: 10.1088/1674-1056/ab41ea

  27. [27]

      F. Curtis, X. Li, T. Rose, et al., J. Chem. Theory Comput. 14 (2018) 2246–2264. doi: 10.1021/acs.jctc.7b01152

  28. [28]

      I. Bier, D. O'Connor, Y.T. Hsieh, et al., CrystEngComm. 23 (2021) 6023–6038. doi: 10.1039/d1ce00745a

  29. [29]

      J. Hu, W. Yang, R. Dong, et al., CrystEngComm. 23 (2021) 1765–1776. doi: 10.1039/d0ce01714k

  30. [30]

      W. Yang, E.M. Dilanga Siriwardane, J. Hu, et al., J. Chem. Phys.126 (2022) 640–647. doi: 10.1021/acs.jpca.1c07170

  31. [31]

      M. Gastegger, L. Schwiedrzik, M. Bittermann, et al., J. Chem. Phys. 148 (2018) 241709.

  32. [32]

      B. Parsaeifard, S. Goedecker, J. Chem. Phys. 156 (2022) 034302.

  33. [33]

      P.T. Różański, M. Zieliński, Comput. Phys. Commun. 287 (2023) 108693.

  34. [34]

      E. Bormashenko, I. Legchenkova, M. Frenkel, et al., Entropy 25 (2023) 92. doi: 10.3390/e25010092

  35. [35]

      L.M. Antunes, R. Grau-Crespo, K.T. Butler, et al., Npj Comput. Mater. 8 (2022) 44.

  36. [36]

      K.T. Schütt, H. Glawe, F. Brockherde, et al., Phys. Rev. B 89 (2014) 205118. doi: 10.1103/PhysRevB.89.205118

  37. [37]

      J. Hu, Y. Zhao, Q. Li, et al., ACS Omega 8 (2023) 26170–26179. doi: 10.1021/acsomega.3c02115

  38. [38]

      G.G.C. Peterson, J. Brgoch, J. Phys. Energy 3 (2021) 022002. doi: 10.1088/2515-7655/abe425

  39. [39]

      C. Chen, J. Zheng, C. Chu, et al., Chin. Chem. Lett. (2024) 109739.

  40. [40]

      G.K. Ozerov, D.S. Bezrukov, A.A. Buchachenko, et al., Low Temp. Phys. 45 (2019) 301–309. doi: 10.1063/1.5090045

• 

  Figure 1  Workflow of stable crystal structure prediction based on deep learning and multi-objective optimization.

  下载: 全尺寸图片 幻灯片
  

  Figure 2  Representation of the crystal BPO4. (a) Crystal parameter table, (b) crystal structure diagram, (c) crystal contact map.

  下载: 全尺寸图片 幻灯片
  

  Figure 3  NaCl crystal contact map analysis. (a) Stable NaCl crystals and their corresponding crystal contact map in the material Project database, (b-d) NaCl crystals and their corresponding crystal contact map predicted by the method in this article.

  下载: 全尺寸图片 幻灯片
  

  Figure 4  The stability analysis of NaCl crystal structures. (a) The optimal NaCl crystal structures obtained through three different objective functions. The convex hulls of NaCl crystals by (b) f = ΔH and (d) f = ΔH + V². The phonon spectra of NaCl crystals by (c) f = ΔH and (e) f = ΔH + V².

  下载: 全尺寸图片 幻灯片

  Table 1.  Properties of optimal NaCl searched by three objective functions.

  Compounds	Objective function	Lattice parameters	Lattice parameters (structure optimization)	Free energy TOTEN (eV)	Formation energy (eV/atom)
  Na4Cl4	f=ΔH	a = 6.45933 Å α = 90.0000°	a = 6.14466 Å α = 90.0000°	27.79312767	-1.88861
  		b = 6.45933 Å β = 90.0000°	b = 6.14466 Å β = 90.0000°
  		c = 6.45933 Å γ = 90.0000°	c = 6.14466 Å γ = 90.0000°
  	f=ΔH+|V|	a = 6.45317 Å α = 90.0000°	a = 6.14375 Å α = 90.0000°	27.79299033	-1.888593
  		b = 6.45317 Å β = 90.0000°	b = 6.14375 Å β = 90.0000°
  		c = 6.45317 Å γ = 90.0000°	c = 6.14375 Å γ = 90.0000°
  	f=ΔH+V2	a = 5.67849 Å α = 90.0000°	a = 5.67849 Å α = 90.0000°	28.72775854	-2.005439
  		b = 5.67849 Å β = 90.0000°	b = 5.67849 Å β = 90.0000°
  		c = 5.67849 Å γ = 90.0000°	c = 5.67849 Å γ = 90.0000°

  下载: 导出CSV
•