Stable structures of Al510–800 clusters and lowest energy sequence of truncated octahedral Al clusters up to 10,000 atoms

Chemical Physics 405 (2012) 100–106

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Stable structures of Al510–800 clusters and lowest energy sequence of truncated octahedral Al clusters up to 10,000 atoms Xia Wu a,b,⇑, Chengdong He a a

School of Chemistry and Chemical Engineering, Anqing Normal University, Anqing 246011, PR China Anhui Provincial Laboratory of Optoelectronic and Magnetism Functional Materials, School of Chemistry and Chemical Engineering, Anqing Normal University, Anqing 246011, PR China b

a r t i c l e

i n f o

Article history: Received 15 December 2011 In final form 28 June 2012 Available online 6 July 2012 Keywords: Aluminum clusters Geometry optimization Truncated octahedron Growth sequence

a b s t r a c t The stable structures of Al510–800 clusters are obtained using dynamic lattice searching with constructed cores (DLSc) method by the NP-B potential. According to the structural growth rule, octahedra and truncated octahedra (TO) configurations are adopted as the inner cores in DLSc method. The results show that in the optimized structures two complete TO structures are found at Al586 and Al711. Furthermore, Al510–800 clusters adopt TO growth pattern on complete TOs at Al405, Al586, and Al711, and the configurations of the surface atoms are investigated. On the other hand, Al clusters with complete TO motifs are studied up to the size 10,000 by the geometrical construction method. The structural characteristics of complete TOs are denoted by the term ‘‘family’’, and the growth sequence of Al clusters is investigated. The lowest energy sequence of complete TOs is proposed. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction Nanoclusters have been widely studied in recent years because of their special electrical, optical, magnetic, and surface properties [1–4]. Aluminum clusters have attracted great interest since aluminum is inexpensive as a light metal, and has been broadly applied in industry. Many interesting phenomena for these nanoclusters have also been reported in Al nanoclusters [5–8]. Furthermore, their chemical and physical properties strongly depend on their sizes (i.e., structures). For instance, aluminum clusters have sizeselective reactivity for the dissociative chemisorption of water [9]. The diverse melting behaviors of the intermediate-size AlN clusters with N around 55 are related to the competition between clusters with different degree of symmetries [10]. The stable geometrical structures and the dependence of structures on cluster sizes are, thus, significant issues, and the investigation of nanoclusters with up to a few hundred atoms is interesting and challenging. The adsorption or reactions of aluminum clusters with small molecules often depend on cluster sizes. For example, aluminum nanowires and aluminum face-centered cubic (fcc) clusters with different sizes could be considered as promising candidates for fuel cell hydrogen storage devices [11]. The most stable structure of Al13H was the hollow hexagonal close-packed (hcp), however, H energetically preferred the hollow fcc site for the icosahedral ⇑ Corresponding author at: School of Chemistry and Chemical Engineering, Anqing Normal University, Anqing 246011, PR China. Tel./fax: +86 556 550 0090. E-mail address: [email protected] (X. Wu). 0301-0104/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.chemphys.2012.06.015

surface [12]. Moreover, Al50 was taken as a superatom to study the electronic structure of Al50(C5(CH3)5)12 complex [13]. Therefore, the stable structures and growth sequence of aluminum clusters were investigated [14]. In our previous work, Al63–500 clusters were identified as truncated octahedra (TO) except for five decahedral structures at Al64, Al72, Al74, Al76, and Al101, four stacking fault face-centered cubic structures at Al91, Al99, Al129, and Al135, and one icosahedron at Al147 [15]. Furthermore, the growth pattern of TO structures in aluminum clusters is found to be based on eight complete TOs at Al38, Al79, Al116, Al140, Al201, Al260, Al314, and Al405 [15,16]. According to Wulff’s construction, a metal nanocrystal with a face-centered cubic structure, e.g., octahedron, tends to be evolved into a truncated octahedral configuration in attempt to minimize the total surface energy [17]. Furthermore, octahedral structures are classified into uncentered regular octahedron [14], e.g., a 146-atom configuration plotted in Fig. 1a, and centered regular octahedron, e.g., a 231-atom configuration plotted in Fig. 2a. If m subshells atoms from each vertex of a regular n-shell octahedron are truncated, several structures with different motifs will be constructed. For example, truncated octahedron, truncated cube, and face-centered cube are obtained from an uncentered octahedron as shown in Fig. 1, and there is no atom at the center of the 14-atom face-centered cube in Fig. 1d. However, from Fig. 2, it is clear that, truncated octahedron, cuboctahedron, and cube are generated, and there is an atom at the center of the 13-atom cuboctahedron in Fig. 2e as plotted in Fig. 2a. Therefore, it can be found that Al clusters are grown based on different families of complete TOs,

X. Wu, C. He / Chemical Physics 405 (2012) 100–106

101

Fig. 1. Configurations truncated from a regular uncentered octahedron: (a) 146-atom octahedron, (b) 116-atom truncated octahedron, (c) 62-atom truncated cube, and (d) 14-atom face-centered cube.

Fig. 2. Configurations truncated from a regular centered octahedron: (a) 231-atom octahedron, (b) 201-atom truncated octahedron, (c) 147-atom cuboctahedron, (d) 63-atom cube, and (e) 13-atom cuboctahedron.

and the fact intrigues us about the question of what roles of complete TO families are in the lowest energy sequence. In this study, the stable structures of Al clusters with 510–800 atoms are optimized using dynamic lattice searching with constructed inner cores (DLSc) method by the NP-B function [18]. With the results, the structural characteristics of these structures are studied, and the structures with complete TO motifs are investigated for understanding the growth pattern. Furthermore, to study the lowest energy sequence of complete TO, all complete TOs within 10,000 atoms are constructed, and investigated systematically. Results show that the lowest energy sequence of completed TOs is proposed as 38, 79, 116, 140, 201, 260, 314, 405, 586, 711, 976, 1289, 1504, 1925, 2190, 2406, 2735, 3054, 3348, 3739, 4500, 4957, 5341, 5882, 6266 6560, 7573, 8894, and 9879 in aluminum clusters. 2. Method 2.1. Dynamic lattice searching with constructed core method Taking the advantage of the modeling strategy and stochastic optimization, a dynamic lattice searching (DLS) method was

proposed for fast optimization of LJ clusters up to 500 atoms [19], Ag13–120 clusters [20], MorseN (N 6 210) clusters [21], (C60)N (N 6 150) clusters [22]. The basic steps of the method can be summarized as an initialization step, i.e., the random generation and local minimization of a structure, and the repetition of dynamic lattice (DL) construction and DL searching. DL construction builds DL around a starting structure in accordance with the fact that just specific positions will be located after local minimization (LM) [23] when one atom is added to a fixed cluster [24]. DL searching operation finds candidates with lower energy by moving the atoms with higher energy to the DL candidates with a lower energy. If the best one of the locally minimized candidates has lower energy than the starting structure, it will be taken as the starting structure for the next generation. Otherwise, the optimization will stop and the current structure will be taken as the result of the run. Because one DLS run cannot ensure the finding of the global minima, the procedure should run Nrun times. In order to reduce the searching space, a strategy by fixing the inner atoms was proposed, named as DLSc [25], because the structure of large clusters is grown shell by shell, and the inner shell atoms are usually located with a definite configuration. For example, in large LJ clusters, the inner core of the clusters with Marks

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decahedral and Mackay icosahedral motifs are Ino’s decahedron and Mackay icosahedron, respectively. The basic steps of the DLS are kept in DLSc. The only difference between DLSc and DLS is their starting structures. In DLSc method, instead of generating the starting structure randomly as done in DLS, an inner core was used to generate the starting structure. To investigate the efficiency of DLSc in optimization of large clusters, it was further applied to the global optimization of LJ clusters with 660 6 N 6 670 [25]. DLSc was used to optimize the structures of the silver clusters from Ag120 to Ag310 [24,26] and aluminum clusters up to 500 atoms [15,16]. In this study, for optimization of the clusters with 510–800 atoms, the inner cores with fcc configurations are used in DLSc method based on the knowledge of the TO structural motifs in aluminum clusters [15]. The final structure was determined by selecting the structure with the lowest energy from the results of the independent runs of DLSc. On the other hand, NP-B potential energy function is used for describing the interaction between atoms of aluminum clusters. The total energy of a system with N atoms is given by

U¼

N X X Fðqi Þ þ /ðr ij Þ i

ð1Þ

i>j

where F(qi) is a many-body interaction that depends on the local electron density at atom i, and /(rij) is an effective two-body interaction. Parameters were described in detail in Ref. [18]. 2.2. Definition of complete TO family Fig. 3 illustrates the structural transformation from regular octahedron to complete truncated octahedron. In the figure, the parameter n is used to define a regular octahedron. For the octahedral structures, the atom number (N = (2  (n + 1)3 + (n + 1))/3) is 6, 19, . . ., 14644 when the number of shells (n) is 1, 2, . . ., 27. The parameters n and m are used to define complete truncated octahedron, and the clusters with the same m values belong to a ‘‘family’’ [27]. By truncating m subshells atoms from each vertex of a regular n-shell octahedron, a complete TO structure with (m + 1)  (m + 1)-atom (1 0 0) face is formed with the remaining atoms. The size of the corresponding complete TO is calculated P 2 by N  6  m i¼1 i , which is listed in Table 1. For instance, 116atom complete TO in Fig. 1 with parameters n = 5 and m = 2

belongs to family 2, and its (1 0 0) face has nine atoms. It should be noted that by truncating truncated octahedron, truncated cube, face-centered cube, cuboctahedron, and cube can be constructed as described above. However, Table 1 just lists all possible complete TOs with sizes N between 20 and 10,000, and other motifs are not considered due to energetically less favorable. In addition, if n is odd, both the regular octahedron and corresponding complete TO are uncentered; otherwise, the center is occupied by an atom, but it has nothing to do with the value of m. From the table, it can be seen that the optimized complete TO structures in Refs. [15,16] belong to different families, e.g., Al38, Al79, and Al140 belong to family 1, Al116, Al201, and Al314 belong to family 2, and both Al260 and Al405 belong to family 3. 3. Results and discussion 3.1. Stable structures of Al510–711 clusters Clusters with tens of atoms and possibly taking a complete TO motif (586 and 711) within the sizes from 510 to 711 are optimized, and the optimized structures are plotted in Fig. 4. For clarity, atoms in the inner cores are shown with dark spheres, and the surface atoms are represented with light spheres in the figure. From Fig. 4, two complete TOs can be found at Al586 and Al711 in the size range of 510–711, respectively, which gives proof of the possible complete TOs predicted as listed in Table 1. In the table, Al586 and Al711 belong to families 3 and 4, respectively. Furthermore, for the sizes smaller than 586, Al405 (family 3) was reported to be the largest complete TO cluster in our previous work [16], the center of which is occupied by an atom. However, it is found that the extra atoms in Al586 over Al405 are located on the five (1 0 0) faces and four (1 1 1) faces, with two layers added on one (1 0 0) face, Al586 thus becomes an uncentered cluster. Further investigation shows that Al510–580 clusters adopt a growth on the surface of the complete TO Al405. To quantitatively analyze the growth rule of the surface atoms, the number of (1 0 0) and (1 1 1) faces on which extra atoms grow with the increase of cluster size, denoted by N(100) and N(111), is given in Table 2. For example, both N(100) and N(111) are 3 for Al510 cluster. At Al520, the extra atoms are added to form another (100) face. Then from Al520 to Al540, clusters prefer forming large number of (111) faces. For Al550–570, N(100) and N(111) are equal. At Al580, one more (111) face is grown with the extra atoms. A complete TO is finally formed at Al586. For clusters of Al590710, the growth is based on the complete TO Al586, except for complete TO at Al711. From Table 2, it can be seen that in Al590–670, the difference between N(100) and N(111) alternatively changes from +1 to 1 then to 0. It can be explained by the reason that, in this size range, all the extra atoms prefer to be located on one face regardless of (1 0 0) or (1 1 1). For example, the extra atoms of Al590 prefer to grow on one (1 0 0) face. However, when the number of extra atoms exceeds the site number of a (1 0 0) face but is less than the site number of a (1 1 1) face, the latter will be occupied, e.g., Al600. For the clusters of Al680–710, more atoms are added to the (1 1 1) faces, because of larger N(111) listed in the table. At Al711, another complete TO is formed. However, the optimization of the structure requires abundant runs of the DLSc method, and it costs much time for so large size. It can thus be deduced that it needs more calculations to optimize the larger size clusters, which may be on the basis of Al586 or Al711. 3.2. Lowest energy sequence of complete truncated octahedra

Fig. 3. Construction of complete truncated octahedron by truncating m subshells atoms from each vertex of a regular n-shell octahedron.

For Al clusters with sizes from 20 to 10,000, thirteen complete TO families in Table 1 are investigated systematically by above

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X. Wu, C. He / Chemical Physics 405 (2012) 100–106 Table 1 Energies of complete truncated octahedral sequence of families 1–13 of aluminum clusters. N Family 1 38 79 140 225 338 483 664 885 1150 1463 1828 2249 2730 3275 3888 4573 5334 7100 9218 Family 6 1288 1709 2190 2735 3348 4033 4794 5635 6560 7573 8678 9879 Family 9 3630 4471 5396 6409 7514 8715 Family 12 7826 9231 Family 13 9730

n

m

E (eV)

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 21 23

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

104.117 228.273 417.420 685.265 1045.519 1511.894 2098.099 2817.844 3684.841 4712.799 5915.428 7306.439 8899.542 10708.448 12746.866 15028.507 17567.082 23442.408 30492.016

13 14 15 16 17 18 19 20 21 22 23 24

6 6 6 6 6 6 6 6 6 6 6 6

4146.681 5537.665 7130.761 8939.665 10978.083 13259.724 15798.298 18607.516 21701.088 25092.724 28782.222 32811.253

19 20 21 22 23 24

9 9 9 9 9 9

11907.934 14717.108 17810.663 21154.971 24844.840 28859.605

25 26

12 12

25821.349 30542.895

27

13

32217.343

N Family 116 201 314 459 640 861 1126 1439 1804 2225 2706 3251 3864 4549 5310 6151 7076 9194 Family 826 1504 1925 2406 2951 3564 4249 5010 5851 6776 7789 8894 Family 2670 3355 4116 4957 5882 6895 8000 9201 Family 4796 5809 6914 8115 9416

n

m

E (eV)

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 23

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

343.217 611.057 971.315 1437.691 2023.897 2743.643 3610.640 4638.598 5841.228 7232.239 8825.342 10634.248 12672.667 14954.308 17492.883 20302.101 23384.339 30453.758

11 13 14 15 16 17 18 19 20 21 22 23

5 5 5 5 5 5 5 5 5 5 5 5

2631.679 4862.241 6253.252 7846.356 9655.261 11693.680 13975.321 16513.895 19323.113 22416.685 25808.322 29511.733

17 18 19 20 21 22 23 24

8 8 8 8 8 8 8 8

8716.258 10997.860 13536.421 16345.633 19439.202 22764.727 26476.552 30416.786

21 22 23 24 25

10 10 10 10 10

15795.604 19187.191 22828.739 26827.566 31176.505

2

5

8

10

construction method. These structures are locally minimized, and the energies of all the investigated clusters are reported in Table 1. To study the roles of complete TO families playing in the lowest energy sequence, the energetic growth properties of these families are investigated. Fig. 5 gives the comparison between energies of the sequence of 13 families. In the figure, the energies of clusters are plotted as (E  Eoct)/N2/3 vs. N, where E is the energy of a complete TO and Eoct is the fitted energy value of the function a + bN1/3 + cN2/3 + dN, where the parameters a, b, c, and d were obtained by least-squares fit to the binding energies of Al fcc regular octahedra within 10,000 atoms. In Fig. 5, the line lying lower corresponds to the sequence with lower energy. It is clear that no line of family always lies lower than other lines, so no sequence of family is always most favored in energy. This is consistent with the our previous studies [15,16]. At first, in the range 20–100, family 1 has relatively lower energy, and corresponding TO Al clusters grow based on the complete TO of Al38 and Al79, respectively, as discussed in Ref [15]. For the larger sizes from 100 to 314, it can be clearly seen that the energies of family 2 become lower than that of family 1. However, the point of Al140 proved to be TO is slightly on the upper of curve family 2. This has been explained by the reason that the four-atom

N Family 260 405 586 807 1072 1385 1750 2171 2652 3197 3810 4495 5256 6097 7022 8035 9140 Family 490 976 1654 2075 2556 3101 3714 4399 5160 6001 6926 7939 9044 Family 1896 2441 3054 3739 4500 5341 6266 7279 8384 9585 Family 6188 7389 8690

n

m

E (eV)

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

798.496 1264.861 1851.068 2570.815 3437.813 4465.771 5668.400 7059.412 8652.515 10461.421 12499.839 14781.481 17320.055 20129.274 23222.846 26614.483 30317.895

9 11 13 14 15 16 17 18 19 20 21 22 23

4 4 4 4 4 4 4 4 4 4 4 4 4

1538.467 3125.193 5355.782 6746.794 8339.897 10148.803 12187.221 14468.862 17007.437 19816.655 22910.227 26301.864 30005.276

15 16 17 18 19 20 21 22 23 24

7 7 7 7 7 7 7 7 7 7

6152.024 7960.896 9999.304 12280.942 14819.515 17628.732 20722.304 24072.067 27781.600 31788.764

23 24 25

11 11 11

20447.818 24410.154 28760.827

3

4

7

11

(1 0 0) faces in Al140 are less favorable than the nine-atom (1 0 0) faces in Al116. Furthermore, the point of 260 is close to the curve of family 2, and it belongs to the sequence. From 320 to 600, the energies of family 3 are lower than those of families 1 and 2. In the size range 600–1000, family 4 has the lowest energies. For the larger sizes from 1000 to 2000, the energies of family 5 are lower than others. In the range 2000–3500, family 6 has the lowest energies, and the points of 2406 (family 5) and 3054 (family 7) are close to the curve. In the range 3500–6300, family 7 is favorable, but family 8 (the point 4957 and 5882) has a competition as shown in the figure. For the larger sizes of 6300–10,000, family 6 is more favorable, with the exception of the point at 8894 in family 5. On the basis of the above analysis, the energetically most stable sequence of complete TO in the range of 20–10,000 can be obtained. The favorable sequence is uncentered Al38 and centered Al79 in family 1, uncentered Al116 (family 2), uncentered Al140 (family 1), centered Al201 (family 2), uncentered Al260 (family 3), uncentered Al314 (family 2), centered Al405 and uncentered Al586 in family 3, centered Al711, uncentered Al976, and centered Al1289 in family 4, uncentered Al1504 and centered Al1925 in family 5, uncentered Al2190 (family 6), uncentered Al2406 (family 5), centered Al2735

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X. Wu, C. He / Chemical Physics 405 (2012) 100–106

Fig. 4. Putative global minimal structures of Al510–711 clusters. Inner core atoms and surface atoms are shown with dark and light spheres, respectively.

(family 6), uncentered Al3054 in family 7, uncentered Al3348 (family 6), centered Al3739 and uncentered Al4500 (family 7), centered Al4957 (family 8), centered Al5341 (family 7), uncentered Al5882 (family 8), uncentered Al6266 (family 7), uncentered Al6560 (family 6), centered Al7573 (family 6), uncentered Al8894 (family 5), and centered Al9879 (family 6). 3.3. Stable structures of Al720–800 clusters on Al711 As discussed above, two complete TOs can be found at Al586 and Al711 in the size range of 510–800. Therefore, the inner cores with

586-atom and 711-atom complete TOs are used in DLSc method to locate the stable structures of Al720–800 clusters, and the lowest energy structures are selected as the global minima. The optimized structures are plotted in Fig. 6. In the figure, the inner part constructing the complete TO is shown with dark spheres, and the extra atoms are represented with light spheres. From Fig. 6, it is clear that all clusters grow based on Al711 represented with dark spheres. The extra atoms in Al720 by light spheres are added on one (1 0 0) face of Al711. At Al730 and Al740, the extra atoms are located on two (1 0 0) faces. However, at Al750, the extra atoms originally located on one (1 0 0) face of

X. Wu, C. He / Chemical Physics 405 (2012) 100–106

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Table 2 The number of (100) and (111) faces, N(100) and N(111), in Al510–711 clusters. N

N(100) a

510 520a 530a 540a 550a 560a 570a 580a 586 590b 600b 610b a,b

3 4 2 3 3 4 4 4 0 1 0 0

N(111) 3 3 4 4 3 4 4 5 0 0 1 1

N b

620 630b 640b 650b 660b 670b 680b 690b 700b 710b 711

N(100)

N(111)

1 2 1 1 2 2 1 2 2 1 0

1 1 2 2 2 2 3 3 3 3 0

Clusters grow on the basis of Al405 and Al586, respectively.

Fig. 6. Putative global minimal structures of Al720–800 clusters. Inner core atoms and surface atoms are shown with dark and light spheres, respectively.

Fig. 5. Energetic comparison of the complete truncated octahedral families 1–13. N is the number of atoms, E is the energy of the cluster, and Eoct is the fitting energy of Al octahedral clusters. Eoct = 17.8432 + 9.7650N1/3 + 1.1008N2/3  3.3825N. Labeled numbers are the possible sizes of complete truncated octahedra in Al clusters within 20–10,000 atoms.

Al740 move to a (1 1 1) face with the new extra atoms. With the increase of sizes, at Al760 and Al770, extra atoms occupy one (1 0 0) face, making N(100) equal to two. Then at Al780, one more (1 1 1) face is filled with atoms originally on one (1 0 0) face of Al770 and the new extra atoms as in Al750. For larger clusters, at Al790 and Al800, extra atoms are added on one (1 0 0) face. It should be mentioned that for the optimization of Al720–800 clusters, the DLSc method with 711-atom TO inner core is more efficient than that with 586-atom TO core. This may be explained from three aspects. At first, a larger size inner core corresponds to a smaller searching space. Apparently, the extra atoms on the surface of the 711-atom TO core are easily to be arranged during the optimization. Second, the interior of the 586- (uncentered) and 711-atom (centered) complete TOs is different. As discussed above, Al710 is based on the 586-atom complete TO, and from Al710 to Al711 clusters, their configurations are rearranged [28] from uncentered to centered as shown in Fig. 4. In general, for a small size cluster, the atomic rearrangement is easily realized by global optimization algorithms, such as genetic algorithm (GA) [29,30], basin hopping method (BH) [31], and simulated annealing (SA) algorithm [32]. However, for a large size cluster, to generate the configurational rearrangement from an uncentered 586-atom complete TO to a centered 711-atom complete TO structure in Al720–800, many atoms on the surface or in the interior should be moved and rearranged. Therefore, 711-atom TO inner core is used for the location of Al720–800 clusters because of their structural growth based on 711-atom complete TO. Third, the surface atoms with high energies are mainly moved in our DLSc algorithm, and it

costs much time to rearrange the interior configuration by moving the surface atoms for so large clusters. Moreover, the 25-atom (100) face of 711-atom complete TO (n = 10, m = 4) can be formed by further truncating one subshell from 807-atom complete TO (n = 10, m = 3) but not 586-atom complete TO (n = 9, m = 3) as listed in Table 1. In fact, if a 586-atom complete TO core is used, one of its 16-atom (100) faces is truncated, and the truncated atoms and the extra atoms are added to other 16-atom (100) faces forming 25atom (100) faces during the optimization. Therefore, the difficulty of the optimization depends on the configuration of the inner core. 4. Conclusions The structures of aluminum clusters from Al510 to Al800 were optimized with DLSc method. The optimized structures took the form of truncated octahedra (TO) including two complete TOs at Al586 and Al711, and the structures of Al510–580, Al590–710, and Al720–800 were found to grow based on the complete TOs of Al405, Al586, and Al711, respectively. Furthermore, the lowest energy sequence of complete TOs in Al clusters within 10,000 atoms was investigated. Results showed that previously optimized complete TOs could be found in the sequence, and the new lowest energy growth sequence was obtained. Moreover, no family of complete TO was found to be dominant in the sequence. It might be helpful for the theoretical guidance of Al clusters and experimental validation. Acknowledgements This study is supported by National Natural Science Foundation of China (NNSFC) (Nos. 21171008 and 20901004). The authors thank X.G. Shao for a grant of DLSc program from Nankai University. The work was carried out at National Supercomputer Center in Tianjin, China and the calculations were performed on TianHe-1(A).

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