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1,3,5-trinitro-1,3,5-triazacyclohexane (RDX) or hexogen, a high-insensitivity explosive, the accurately description of its energy and properties is of fundamental significance in the sense of security and application. Based on the machine learning method, high-dimensional neural network is used to construct potential function of RDX crystal. In order to acquire enough data in neural network learning, based on the four known crystal phases of RDX, the structural global search is performed under different spatial groups to obtain 15199 structure databases. Here in this study, we use nearby atomic environment to build 72 different basis functions as input neurons, in which the 72 different basis functions represent the interaction with nearby atoms for each type of element. Among them, 90% data are randomly set as training set, and the remaining 10% data are taken as test set. To obtain the better training effect, 9 different neural network structures carry out 2000 step iterations at most, thereby the 30-30-10 hidden layer structure has the lower root mean square error (RMSE) after the 1847 iterations compared with the energies from first-principles calculations. Thus, the potential function fitted by 30-30-10 hidden layer network is chosen in subsequent calculations. This constructed potential function can reproduce the first-principles results of test set well, with the RMSE of 59.2 meV/atom for binding energy and 7.17 eV/Å for atomic force. Especially, the RMSE of the four known RDX crystal phases from 1 atm to 6 GPa are 10.0 meV/atom and 1.11 eV/Å for binding energy and atomic force, respectively, indicating that the potential function has a better description of the known structures. Furthermore, we also propose four additional RDX crystal phases with lower enthalpy, which may be alternative crystal phases undetermined in experiment. In addition, based on molecular dynamics simulation with this potential function, the α-phase RDX crystal can stay stable for a few ps, further proving the applicability of our constructed potential function.
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Keywords:
- energetic material /
- neural networks /
- potential function /
- molecular dynamics
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Wang Z S 2006 Introduction to Energetic Material (Harbin: Harbin Institute of Technology Press) pp4−8 (in Chinese)
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Google Scholar
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Google Scholar
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Song H, Li H, Zhang P, Yang Y, Huang F 2018 Chin. J. Energ. Mater. 26 1006
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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图 1 高维神经网络结构示意图, C代表原子坐标, G代表基底函数, H代表隐藏层神经元, E代表原子能量, 下角标1, 2, ···, n为原子序号, ET为体系总能量
Fig. 1. Structure of high-dimensional neural network. C, G, H, and E represent coordinates of atom, basis functions, hidden layer neurons, and energy of atom, respectively. Subscripts, 1, 2, ···, n are the serial numbers of atoms, and ET is the total energy of the system.
图 3 (a) 9种隐藏层结构在400代之后测试集最低RMSE随迭代步数变化示意图; (b) 30-30-10隐藏层网络结构在1500代之后训练集和测试集RMSE随迭代步数变化示意图
Fig. 3. (a) Diagram of test set lowest RMSEs variation along with training iteratons of nine types hidden layer neural structures after 400 iterations; (b) diagram of training and test sets RMSEs variation along with training iteratons of 30-30-10 hidden layer neural structures after 1500 iterations.
图 4 (a) 训练集(黑色叉)和测试集(红色十字)所有结构第一性原理计算形成能和机器学习计算形成能对应关系图; (b) 训练集(黑色叉)和测试集(红色十字)所有结构第一性原理计算原子受力和机器学习计算原子受力对应关系示意图
Fig. 4. (a) Correlation of machine learning binding energies with the corresponding ab initio reference energies for all structures in the training (black skew crosses) and testing (red crosses) sets; (b) correlation of machine learning atomic forces with the corresponding ab initio reference forces for all structures in the training (black skew crosses) and testing (red crosses) sets.
图 5 0 K下4种已知RDX晶型在标准大气压到6 GPa压强下机器学习计算结合能的误差
Fig. 5. Errors of machine learning binding energies of four known RDX crystals from 1 atm to 6 GPa at 0 K.
图 6 (a) 4种RDX晶型结构示意图; (b) 7—10 GPa下4种晶型第一性原理计算(黑色方块)和机器学习势函数计算(红色圆圈)的焓值
Fig. 6. (a) Structures of four RDX crystals; (b) enthalpies from 7 to 10 GPa calculated by ab initio (black block) and machine learning potential (red circle).
图 7 (a) α相RDX 2 × 2 × 2晶胞在NVT系综下分子动力学模拟温度随时间变化图; (b) 压强随时间变化图
Fig. 7. Variations in time of the temperature (a) and pressure (b) in the NVT ensemble for 2 × 2 × 2 α-RDX crystal.
No. 1—4 5—8 9—12 η/Å–2 0.003214 0.214264 1.428426 
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表 2 10 × 6个G4型角向基函数((1b)式)中η, λ, ζ取值
Table 2. η, λ, and ζ of 10 × 6 G4 type angular basis functions (Eq. (1b)).
No. 13—22 23—32 33—42 43—52 53—62 63—72 η/10-4 Å–2 3.57 3.57 3.57 3.57 3.57 3.57 λ –1.0 1.0 –1.0 1.0 –1.0 1.0 ζ 1.0 1.0 2.0 2.0 4.0 4.0
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表 3 训练集与测试集机器学习计算形成能和原子受力与第一性原理计算比较MAE和RMSE
Table 3. MAE and RMSE of machine learning binding energies and atomic forces corresponding ab initio reference energies and forces in the training and test sets.
Energy/meV·atom–1 Force/eV·Å–1 MAE RMSE MAE RMSE Training set 29.2 47.1 2.22 9.45 Test set 35.1 59.2 2.24 7.17
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[1] 王泽山 2006 含能材料概论 (哈尔滨: 哈尔滨工业大学出版社) 第4−8页
Wang Z S 2006 Introduction to Energetic Material (Harbin: Harbin Institute of Technology Press) pp4−8 (in Chinese)
[2] Infante-Castillo R, Pacheco-Londono L C, Hernandez-Rivera S P 2010 J. Mol. Struct. 970 51
Google Scholar
[3] Figueroa-Navedo A M, Ruiz-Caballero J L, Pacheco-Londono L C, Hernandez-Rivera S P 2016 Cryst. Growth Des. 16 3631
Google Scholar
[4] Choi C S, Prince E 1972 Acta Crystallogr., Sect. B: Struct. Sci. B 28 2857
[5] Torres P, Mercado L, Cotte I, Hernandez S P, Mina N, Santana A, Chamberlain R T, Lareau R, Castro M E 2004 J. Phys. Chem. B 108 8799
Google Scholar
[6] Millar D I A, Oswald I D H, Francis D J, Marshall W G, Pulham C R, Cumming A S 2009 Chem. Commun. 5 562
[7] Gao C, Yang L, Zeng Y, Wang X, Zhang C, Dai R, Wang Z, Zheng X, Zhang Z 2017 J. Phys. Chem. C 121 17586
Google Scholar
[8] Dreger Z A, Gupta Y M 2010 J. Phys. Chem. A 114 8099
Google Scholar
[9] Sorescu D C, Rice B M 2016 J. Phys. Chem. C 120 19547
Google Scholar
[10] Munday L B, Chung P W, Rice B M, Solares S D 2011 J. Phys. Chem. B 115 4378
Google Scholar
[11] Weingarten N S, Sausa R C 2015 J. Phys. Chem. A 119 9338
Google Scholar
[12] Mathew N, Picu R C 2011 J. Chem. Phys. 135 024510
Google Scholar
[13] Davidson A J, Oswald I D H, Francis D J, Lennie A R, Marshall W G, Millar D I A, Pulham C R, Warren J E, Cumming A S 2008 CrystEngComm 10 162
Google Scholar
[14] Ciezak J A, Jenkins T A 2008 Propellants Explos. Pyrotech. 33 390
Google Scholar
[15] Millar D I A, Oswald I D H, Barry C, Francis D J, Marshall W G, Pulham C R, Cumming A S 2010 Chem. Commun. 46 5662
Google Scholar
[16] Sorescu D C, Rice B M, Thompson D L 1997 J. Phys. Chem. B 101 798
Google Scholar
[17] Sorescu D C, Rice B M, Thompson D L 2000 J. Phys. Chem. B 104 8406
Google Scholar
[18] Guo Y, Thompson D L 1999 J. Phys. Chem. B 103 10599
Google Scholar
[19] Liu H, Zhao J, Ji G, Gong Z, Wei D 2006 Physica B 382 334
Google Scholar
[20] Kohno Y, Ueda K, Imamura A 1996 J. Phys. Chem. 100 4701
Google Scholar
[21] Duan X H, Li W P, Pei C H, Zhou X Q 2013 J. Mol. Model. 19 3893
Google Scholar
[22] Lysne P C, Hardesty D R 1973 J. Chem. Phys. 59 6512
Google Scholar
[23] Strachan A, van Duin A C T, Chakraborty D, Dasgupta S, Goddard W A 2003 Phys. Rev. Lett. 91 098301
Google Scholar
[24] van Duin A C T, Dasgupta S, Lorant F, Goddard W A 2001 J. Phys. Chem. A 105 9396
Google Scholar
[25] Guo F, Cheng X L, Zhang H 2012 J. Phys. Chem. A 116 3514
Google Scholar
[26] Wood M A, van Duin A C T, Strachan A 2014 J. Phys. Chem. A 118 885
Google Scholar
[27] Butler K T, Davies D W, Cartwright H, Isayev O, Walsh A 2018 Nature 559 547
Google Scholar
[28] Blank T B, Brown S D, Calhoun A W, Doren D J 1995 J. Chem. Phys. 103 4129
Google Scholar
[29] Behler J, Parrinello M 2007 Phys. Rev. Lett. 98 146401
Google Scholar
[30] Rupp M, Tkatchenko A, Mueller K R, von Lilienfeld O A 2012 Phys. Rev. Lett. 108 058301
Google Scholar
[31] Bartok A P, Payne M C, Kondor R, Csanyi G 2010 Phys. Rev. Lett. 104 136403
Google Scholar
[32] Vu K, Snyder J C, Li L, Rupp M, Chen B F, Khelif T, Mueller K R, Burke K 2015 Int. J. Quantum Chem. 115 1115
Google Scholar
[33] Hansen K, Montavon G, Biegler F, Fazli S, Rupp M, Scheffler M, von Lilienfeld O A, Tkatchenko A, Mueller K-R 2013 J. Chem. Theory Comput. 9 3404
Google Scholar
[34] Schuett K T, Arbabzadah F, Chmiela S, Mueller K R, Tkatchenko A 2017 Nat. Commun. 8 13890
Google Scholar
[35] Musil F, De S, Yang J, Campbell J E, Day G M, Ceriotti M 2018 Chem. Sci. 9 1289
Google Scholar
[36] Schran C, Behler J, Marx D 2020 J. Chem. Theory Comput. 16 88
Google Scholar
[37] Elton D C, Boukouvalas Z, Butrico M S, Fuge M D, Chung P W 2018 Sci. Rep. 8 9059
Google Scholar
[38] Akkermans R L C, Spenley N A, Robertson S H 2013 Mol. Simul. 39 1153
Google Scholar
[39] Day G M, Motherwell W D S, Ammon H L, Boerrigter S X M, Della Valle R G, Venuti E, Dzyabchenko A, Dunitz J D, Schweizer B, van Eijck B P, Erk P, Facelli J C, Bazterra V E, Ferraro M B, Hofmann D W M, Leusen F J J, Liang C, Pantelides C C, Karamertzanis P G, Price S L, Lewis T C, Nowell H, Torrisi A, Scheraga H A, Arnautova Y A, Schmidt M U, Verwer P 2005 Acta Crystallogr., Sect. B: Struct. Sci. 61 511
Google Scholar
[40] 宋华杰, 李华, 张平, 杨延强, 黄风雷 2018 含能材料 26 1006
Song H, Li H, Zhang P, Yang Y, Huang F 2018 Chin. J. Energ. Mater. 26 1006
[41] Kresse G, Furthmuller J 1996 Phys. Rev. B 54 11169
Google Scholar
[42] Perdew J P, Burke K, Ernzerhof M 1996 Phys. Rev. Lett. 77 3865
Google Scholar
[43] Blochl P E 1994 Phys. Rev. B 50 17953
Google Scholar
[44] Artrith N, Urban A 2016 Comput. Mater. Sci. 114 135
Google Scholar
[45] Artrith N, Urban A, Ceder G 2017 Phys. Rev. B 96 014112
Google Scholar
[46] Behler J 2015 Int. J. Quantum Chem. 115 1032
Google Scholar
[47] Montavon G, Genevive B O, Müller K R (Eds.) 2012 Neural Networks: Tricks of the Trade, Lecture Notes in Computer Science (Ed. 2) (Vol. 7700) (Berlin Heidelberg: Springer-Verlag) p19
[48] Rumelhart D E, Hinton G E, Williams R J 1986 Nature 323 533
Google Scholar
[49] Plimpton S 1995 J. Comput. Phys. 117 1
Google Scholar
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