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Illustrated by the case of the planar clusters, we propose a new method to search the possible stable structures by combining the structural identification and Monte-Carlo tree algorithm. We adopt two kinds of model-potential to describe the interaction between atoms:the pair interaction of Lennard-Jones potential and three-body interaction based on the Lennard-Jones potential. Taking the possible triangular lattice fragment as candidates, we introduce a new nomenclature to distinguish the structures, which can be used for the rapid congruence check. 1) We label the atoms on the triangular lattice according to the distances and the polar angles. where a given triangular structure has a corresponding serial number in the numbered plane. Note that the congruent structures can have a group of possible serial numbers. 2) We consider all the possible symmetrical operations including translation, inversion and rotation, and obtain the smallest one for the unique nomenclature of the structure. In conventional search of magic clusters, the global optimizations are performed for the structures with given number of atoms. Herein, we perform the Monte-Carlo tree search to study the evolution of stable structures with various numbers of atoms. From the structures of given number of atoms, we sample the structures according to their energy with the importance sampling, and then expand the structures to the structures with one more atom, where the congruence check with the nomenclature is adopted to avoid numerous repeated evaluations of candidates. Since the structures various numbers of atoms are correlated with each other, a searching tree will be obtained. In order to prevent the over-expansion of branches, we prove the “tree” according to energy to make the tree asymmetric growth to retain the low energy structure. The width and depth of search is balanced by the control of temperature in the Monte-Carlo tree search. For the candidates with lower energies, we further perform the local optimization to obtain the more stable structures. Our calculations show that the triangular lattice fragments will be more stable under the pair interaction of Lennard-Jones potential, which are in agreement with the previous studies. Under the three body interaction with the specific parameter, the hexagonal lattice fragments will be more stable, which are similar to the configurations of graphene nano-flakes. Combining the congruence check and Monte-Carlo tree search, we provide an effective avenue to screen the possible candidates and obtain the stable structures in a shorter period of time compared with the common global optimizations without the structural identification, which can be extended to search the stable structure for materials by the first-principles calculations.
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Keywords:
- cluster /
- structural identification /
- Monte-Carlo tree /
- global optimization
[1] Baletto F, Ferrando R 2005 Rev. Mod. Phys. 77 371
[2] Gong X F, Wang Y, Ning X J 2008 Chin. Phys. Lett. 25 468
[3] Liu T D, Zheng J W, Shao G F, Fan T E, Wen Y H 2015 Chin. Phys. B 24 33601
[4] Zhang M, Gao Y, Fang H P 2016 Chin. Phys. B 25 13602
[5] de Heer W A 1993 Rev. Mod. Phys. 65 611
[6] Knight W D, Clemenger K, Heer W A D, Saunders W A, Chou M Y, Cohen M L 1984 Phys. Rev. Lett. 52 2141
[7] Honeycutt J D, Andersen H C 1987 J. Phys. Chem. 91 4950
[8] Liu G Q, Zhang Y L, Wang Z X, Wang Y Z, Zhang X X, Zhang W X 2012 Comput. Theor. Chem. 993 118
[9] Li S F, Zhao X J, Xu X S, Gao Y F, Zhang Z Y 2013 Phys. Rev. Lett. 111 115501
[10] Kim S, Hwang S W, Kim M K, Shin D Y, Shin D H, Kim C O, Yang S B, Park J H, Hwang E, Choi S H, Ko G, Sim S, Sone C, Choi H J, Bae S, Hong B H 2012 ACS Nano 6 8203
[11] Dahl J E, Liu S G, Carlson R M K 2003 Science 299 96
[12] Yang X B, Zhao Y J, Xu H, Yakobson B I 2011 Phys. Rev. B 83 205314
[13] Sergeeva A P, Popov I A, Piazza Z A, Li W L, Romanescu C, Wang L S, Boldyrev A I 2014 Acc. Chem. Res. 47 1349
[14] Xu S G, Zhao Y J, Liao J H, Yang X B 2015 J. Chem. Phys. 142 214307
[15] Hartke B 1993 J. Phys. Chem. 97 9973
[16] Wang Y C, L J, Zhu L, Ma Y M 2010 Phys. Rev. B 82 094116
[17] Frontera C, Vives E, Castan T, Planes A 1995 Phys. Rev. B 51 11369
[18] Kresse G, Jurgen H 1993 Phy. Rev. B 47 558
[19] Zhang Y J, Xiao X Y, Li Y Q, Yan Y H 2012 Acta Phys. Sin. 61 093602 (in Chinese)[张英杰,肖绪洋,李永强, 颜云辉2012物理学报61 093602]
[20] Liu T D, Li Z P, Ji Q S, Shao G F, Fan T E, Wen Y H 2017 Acta Phys. Sin. 66 053601 (in Chinese)[刘暾东, 李泽鹏, 季清爽, 邵桂芳, 范天娥, 文玉华2017物理学报 66 053601]
[21] Wu L J, Sui Q T, Zhang D, Zhang L, Qi Y 2015 Acta Phys. Sin. 64 042102 (in Chinese)[吴丽君, 随强涛, 张多, 张林, 祁阳2015物理学报64 042102]
[22] Li P F, Zhang Y G, Lei X L, Pan B C 2013 Acta Phys. Sin. 62 143602 (in Chinese)[李鹏飞, 张艳革, 雷雪玲, 潘必才2013物理学报 62 143602]
[23] L J, Wang Y C, Zhu L, Ma Y M 2012 J. Chem. Phys. 137 084104
[24] Oganov A R, Glass C W 2006 J. Chem. Phys. 124 244704
[25] Solovyov I A, Solovyov A V, Greiner W, Koshelev A, Shutovich A 2003 Phys. Rev. Lett. 90 053401
[26] Swiechowski M, Mandziuk J, Ong Y S 2016 IEEE Trans. Comp. Intel. AI. 8 218
[27] Villar S S, Bowden J, Wason J 2015 Stat. Sci. 30 199
[28] Sasaki Y, de Garis H 2004 Proceedings of the 2003 Congress on Evolutionary Computation Canberra, ACT, Australia, December 8-12, 2003 p886
[29] Yang J, Zhang W Q 2007 Acta Phys. Sin. 56 4017 (in Chinese)[杨炯, 张文清2007物理学报56 4017]
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[1] Baletto F, Ferrando R 2005 Rev. Mod. Phys. 77 371
[2] Gong X F, Wang Y, Ning X J 2008 Chin. Phys. Lett. 25 468
[3] Liu T D, Zheng J W, Shao G F, Fan T E, Wen Y H 2015 Chin. Phys. B 24 33601
[4] Zhang M, Gao Y, Fang H P 2016 Chin. Phys. B 25 13602
[5] de Heer W A 1993 Rev. Mod. Phys. 65 611
[6] Knight W D, Clemenger K, Heer W A D, Saunders W A, Chou M Y, Cohen M L 1984 Phys. Rev. Lett. 52 2141
[7] Honeycutt J D, Andersen H C 1987 J. Phys. Chem. 91 4950
[8] Liu G Q, Zhang Y L, Wang Z X, Wang Y Z, Zhang X X, Zhang W X 2012 Comput. Theor. Chem. 993 118
[9] Li S F, Zhao X J, Xu X S, Gao Y F, Zhang Z Y 2013 Phys. Rev. Lett. 111 115501
[10] Kim S, Hwang S W, Kim M K, Shin D Y, Shin D H, Kim C O, Yang S B, Park J H, Hwang E, Choi S H, Ko G, Sim S, Sone C, Choi H J, Bae S, Hong B H 2012 ACS Nano 6 8203
[11] Dahl J E, Liu S G, Carlson R M K 2003 Science 299 96
[12] Yang X B, Zhao Y J, Xu H, Yakobson B I 2011 Phys. Rev. B 83 205314
[13] Sergeeva A P, Popov I A, Piazza Z A, Li W L, Romanescu C, Wang L S, Boldyrev A I 2014 Acc. Chem. Res. 47 1349
[14] Xu S G, Zhao Y J, Liao J H, Yang X B 2015 J. Chem. Phys. 142 214307
[15] Hartke B 1993 J. Phys. Chem. 97 9973
[16] Wang Y C, L J, Zhu L, Ma Y M 2010 Phys. Rev. B 82 094116
[17] Frontera C, Vives E, Castan T, Planes A 1995 Phys. Rev. B 51 11369
[18] Kresse G, Jurgen H 1993 Phy. Rev. B 47 558
[19] Zhang Y J, Xiao X Y, Li Y Q, Yan Y H 2012 Acta Phys. Sin. 61 093602 (in Chinese)[张英杰,肖绪洋,李永强, 颜云辉2012物理学报61 093602]
[20] Liu T D, Li Z P, Ji Q S, Shao G F, Fan T E, Wen Y H 2017 Acta Phys. Sin. 66 053601 (in Chinese)[刘暾东, 李泽鹏, 季清爽, 邵桂芳, 范天娥, 文玉华2017物理学报 66 053601]
[21] Wu L J, Sui Q T, Zhang D, Zhang L, Qi Y 2015 Acta Phys. Sin. 64 042102 (in Chinese)[吴丽君, 随强涛, 张多, 张林, 祁阳2015物理学报64 042102]
[22] Li P F, Zhang Y G, Lei X L, Pan B C 2013 Acta Phys. Sin. 62 143602 (in Chinese)[李鹏飞, 张艳革, 雷雪玲, 潘必才2013物理学报 62 143602]
[23] L J, Wang Y C, Zhu L, Ma Y M 2012 J. Chem. Phys. 137 084104
[24] Oganov A R, Glass C W 2006 J. Chem. Phys. 124 244704
[25] Solovyov I A, Solovyov A V, Greiner W, Koshelev A, Shutovich A 2003 Phys. Rev. Lett. 90 053401
[26] Swiechowski M, Mandziuk J, Ong Y S 2016 IEEE Trans. Comp. Intel. AI. 8 218
[27] Villar S S, Bowden J, Wason J 2015 Stat. Sci. 30 199
[28] Sasaki Y, de Garis H 2004 Proceedings of the 2003 Congress on Evolutionary Computation Canberra, ACT, Australia, December 8-12, 2003 p886
[29] Yang J, Zhang W Q 2007 Acta Phys. Sin. 56 4017 (in Chinese)[杨炯, 张文清2007物理学报56 4017]
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