First-Principles Studies on the Structural and Electronic Properties of As Clusters

Based on the genetic algorithm (GA) incorporated with density functional theory (DFT) calculations, the structural and electronic properties of neutral and charged arsenic clusters Asn (n = 2–24) are investigated. The size-dependent physical properties of neutral clusters, such as the binding energy, HOMO-LUMO gap, and second difference of cluster energies, are discussed. The supercluster structures based on the As8 unit and As2 bridge are found to be dominant for the larger cluster Asn (n ≥ 8). Furthermore, the possible geometric structures of As28, As38, and As180 are predicted based on the growth pattern.


Introduction
In recent years, due to the fast development of nanotechnology, people are more interested in atomic clusters, which are composed of several to thousands of atoms, molecules, or ions through a physical or chemical bonding force [1]. Clusters can also be regarded as the transitional forms between atoms and bulk, and their fundamental properties depend vitally on the cluster size. Therefore, it is quite meaningful to study the structural and electronic properties of clusters using theoretical research, identifying their potential capacities for numerous applications. Arsenic has been widely applied in many fields such as semiconductors, optoelectronics, and biopharmaceutics [2][3][4][5][6][7][8][9][10][11]. Besides, vanadium doped phosphorus clusters, and pure and doped arsenic clusters have received a large amount of attention from both experimental and theoretical fields in recent years [12][13][14][15][16][17][18][19][20][21][22][23].
Experimentally, the study of As n clusters has focused on small-sized clusters with n ≤ 5 [12][13][14][15][16][17]. For example, Wang et al. [12] utilized high-resolution photoelectron spectroscopy to study the electronic vibration and spin orbit of As 4 . Bennett et al. [13] have measured the appearance potentials and ion translational energies for the As 1 , As 2 , and As 3 ions formed by the dissociative resonance capture of As 4 . Lippa et al. [14] have probed the electron affinities of As n (n = 1-5) using photoelectron spectra in 1998. Brumbach and Rosenblatt [15] have investigated the vibrational modes of As 4 with Raman spectroscopy. Yonezo [16] designed a high-temperature nozzle assembly for gas-electron diffraction to determine the structure of As 4 . Jeffrey et al. [17] measured the ionization potentials (IPs) for As n (n = 1-5) using gas-phase charge-transfer reactions. No experimental data are available for As n with n ≥ 6 right now.
In the theoretical aspect, Zhao et al. [18] and Bai et al. [21] studied the structures, thermochemistry, and electron affinities of As n (n = 1-16) and their anions. Their results showed that the even-numbered neutral As n species are more stable than the odd-numbered clusters, but the even-numbered anionic As n species are less stable than the odd-numbered species. Liang et al. [19] probed the electronic structure and property of neutral and charged arsenic clusters As n (+1,0,−1) (n = 2-8). At the B3LYP/6-311+G(d) theoretical level, Guo [20] investigated the geometries and energies for neutral and charged As n (n = 2-15) clusters, and reported their relative stability, ionization potential, and electron affinity. Baruah et al. [23] using a generalized gradient approximation (GGA) to explore the geometry, vibrational modes, and polarizabilities, as well as the infrared and Raman spectra of fullerene-like arsenic cages with n = 4, 8, 20, 28, 32, 36, and 60. Zhao et al. [22] investigated the structures and electronic properties of As n clusters with even-numbered As n (n = 6-28) using density functional theory (DFT) with the Perdew-Burke-Ernzerhof functional and a doubled numerical basis set with d-polarization functions(PBE/DND) scheme and found that the supercluster structures based on As 4 , As 6 , and As 8 units, and the As 2 bridge were dominant for the larger As n with n ≥ 14. Although many theoretical works have been performed on the As n clusters, all the results are always most of the artificial speculation studies to investigate the structure of clusters in certain symmetries. In this work, we have performed a genetic algorithm (GA) incorporated with density functional theory (DFT) calculations to explore the structures and electronic properties of As n (n = 2-24) neutral and charged clusters. After we have determined the growth pattern of As n clusters, the possible geometric structures of As 28 , As 38 , and As 180 are predicted based on the growth pattern. We also discussed the size-dependent physical properties of neutral clusters such as the binding energy, HOMO-LUMO gap and second difference of cluster energies.

Computational Methodology
In order to search the global minimum structures of As n clusters, we combined a genetic algorithm (GA) simulation with local optimization at the Dmol 3 level [24][25][26][27][28][29]. The fundamental aim in GA is to divide the potential-energy surface (PES) into a number of regions and find the locally stable isomers in each region. In the GA program, we generated 15 As n (n = 3-12) and 20 As n (n = 13-24) initial populations to ensure that we could find the local minimum. Any population can be chosen as parents to generate their child cluster according to a crossover operation. In addition, there was a possibility of a 30% mutation rate for a single parent to produce child alone. The child cluster was optimized with Dmol 3 , and then compared with its parent in energy. The child with lower energy replaced its high-energy parent. The whole process of genetic algorithm with 2000 iterations was to ensure we got the lowest energy structure.
The optimization of As n (n = 2-24) clusters was performed using DFT with the Perdew-Burke-Ernzerhof (PBE) [24] exchange-correlation functional and an all-electron basis set of the double-numerical-plus-d-polarization (DND) type, as implemented in the Dmol 3 [25] package. A self-consistent field calculation kept the accuracy with an energy convergence for 10 −6 a.u., and the forces for 2 × 10 −3 a.u. There was no symmetry restriction for geometry optimization.
For each As n (n = 2-24) cluster, we saved ten energetically lower isomer structures for further electronic structure calculations, which were performed using the Vienna Ab-initio Simulation Package (VASP) codes. The Kohn-Sham equations were solved variationally in a plane wave basis set using the projector-augmented-wave (PAW) method. The exchange-correlation energy was described by the functional of Perdew, Burke, and Ernzerhof (PBE) based on the generalized gradient approximation (GGA). The energy cutoff was set to be 400 eV and the vacuum space was set to be at least 14 Å to separate the interactions between the neighboring slabs. Only the Gamma k-point was used to sample the Brillouin zone for the geometry and electronic structure calculations. All structures were fully relaxed by Gaussian smearing and electronic structure calculations by tetrahedron smearing method until the convergence criteria (with the force less than 0.02 eV/Å and the energy less than 10 −5 eV). Although each structure of As n (n = 2-24) cluster was further optimized by VASP, the energy sequencing of each cluster was basically unchanged.

As n (n = 2-8) Clusters
The bond length of an As 2 cluster (2a in Figure 1) is 2.103 Å through experimental measurement [30]. In our calculation the distance between the As atoms was 2.118 Å, which is closer to the experimental data compared with the 2.142 Å calculated using PBE/DND methods [22].

Asn (n = 2-8) Clusters
The bond length of an As2 cluster (2a in Figure 1) is 2.103 Å through experimental measurement [30]. In our calculation the distance between the As atoms was 2.118 Å, which is closer to the experimental data compared with the 2.142 Å calculated using PBE/DND methods [22].
For the As3 cluster, the energy of the structure with C2v symmetry (3a in Figure 1) was the global minimum. It was an isosceles triangle structure with a top angle of 65.14° and side length of 2.325 Å. It was energetically lower than the linear chain structure with D∞h symmetry (3b in Figure 1), in which structure, the bond length was 2.204 Å. The ground-state structure of As4 with Td symmetry (4a in Figure 1) was a regular tetrahedron, which was consistent with the previous reports [18][19][20]. Its energy was much lower than other isomorphic configurations. The energy of the rectangular structure with D2h symmetry (4b in Figure  1) was 0.605 eV/atom higher than that of the ground state and the chain structure with C2 symmetry (4c in Figure 1) was energetically higher than the rectangular structure.
The lowest energy structure of As5 had C2v symmetry (5a in Figure 1) and it may be considered as adding an atom in the cross section of a dihedral formed by four atoms. It had 0.07 eV/atom less energy than the rectangular pyramid structure with D4h symmetry (5b in Figure 1) and 0.142 eV/atom less than the planar structure with D5h symmetry (5c in Figure 1).
The trigonal prism with D3h symmetry (6a in Figure 1) was the ground-state structure for As6 and was only 0.005 eV/atom lower than the benzvalene type with C2v symmetry (6b in Figure 1) and 0.117 eV/atom lower than the dihedral angle structure of six atoms with C2v symmetry (6c in Figure  1). The side length was 2.522 Å and the edge length was 2.559 Å. Our ground state was consistent with the result calculated by B3LYP/6-311+G(d) [20] or PBE/DND methods [22]. However, the structure 6b in Figure 1 was found to be the lowest energy by Liang [19] using MP2(full)/g-31G(d) methods and Bai [21] using B3LYP/DZP++ methods. As the outer shell structure of As is 3s 2 3p 3 , we think our results are reasonable as the completely three-coordination structure (6a in Figure 1) must be more stable than the structure with two two-coordinations (6b in Figure 1).
In the case of As7, the ground-state structure with C2v symmetry (7a in Figure 1) could be derived from the trigonal prism of As6 by edge-capping with an additional As atom. This low-energy For the As 3 cluster, the energy of the structure with C 2v symmetry (3a in Figure 1) was the global minimum. It was an isosceles triangle structure with a top angle of 65.14 • and side length of 2.325 Å. It was energetically lower than the linear chain structure with D∞h symmetry (3b in Figure 1), in which structure, the bond length was 2.204 Å.
The ground-state structure of As 4 with T d symmetry (4a in Figure 1) was a regular tetrahedron, which was consistent with the previous reports [18][19][20]. Its energy was much lower than other isomorphic configurations. The energy of the rectangular structure with D 2h symmetry (4b in Figure 1) was 0.605 eV/atom higher than that of the ground state and the chain structure with C 2 symmetry (4c in Figure 1) was energetically higher than the rectangular structure.
The lowest energy structure of As 5 had C 2v symmetry (5a in Figure 1) and it may be considered as adding an atom in the cross section of a dihedral formed by four atoms. It had 0.07 eV/atom less energy than the rectangular pyramid structure with D 4h symmetry (5b in Figure 1) and 0.142 eV/atom less than the planar structure with D 5h symmetry (5c in Figure 1).
The trigonal prism with D 3h symmetry (6a in Figure 1) was the ground-state structure for As 6 and was only 0.005 eV/atom lower than the benzvalene type with C 2v symmetry (6b in Figure 1) and 0.117 eV/atom lower than the dihedral angle structure of six atoms with C 2v symmetry (6c in Figure 1). The side length was 2.522 Å and the edge length was 2.559 Å. Our ground state was consistent with the result calculated by B3LYP/6-311+G(d) [20] or PBE/DND methods [22]. However, the structure 6b in Figure 1 was found to be the lowest energy by Liang [19] using MP2(full)/g-31G(d) methods and Bai [21] using B3LYP/DZP++ methods. As the outer shell structure of As is 3s 2 3p 3 , we think our results are reasonable as the completely three-coordination structure (6a in Figure 1) must be more stable than the structure with two two-coordinations (6b in Figure 1).
In the case of As 7 , the ground-state structure with C 2 v symmetry (7a in Figure 1) could be derived from the trigonal prism of As 6 by edge-capping with an additional As atom. This low-energy structure is also predicted in Refs. [19][20][21]. Its energy was lower than the structure with Cs symmetry (7b in Figure 1) by 0.008 eV/atom and the structure with Cs symmetry (7c in Figure 1) by 0.053 eV/atom.
The wedge-like structure that looks like a cage with C 2v symmetry was obtained as the lowest energy structure for As 8 , as seen from 8a in Figure 1. It was energetically lower than the structure with C 2v symmetry (8b in Figure 1) by 0.038 eV/atom and the structure with Cs symmetry (8c in Figure 1) by 0.048 eV/atom. We also checked the cage structure with O h symmetry cut from the bulk phase reported by Baruah [23], and we found that it was 0.64 eV energy higher than our ground state structure.
After analysis of the ground structures of As n (n = 2-8) clusters, we find that As 2 was a one-dimensional bridge, As 3 was a two-dimensional isosceles triangle and As 4 became a three-dimensional tetrahedron. When n was larger than 3, the two-dimensional cluster structure, such as 4b and 5c in Figure 1, sorts more and more backward energetically. We can conclude that the structure of small As clusters tends towards a three-dimensional cage structure and it was not stable for a 2-D planar structure.

As n (n = 9-18) Clusters
The ground-state structure of As 9 with C s symmetry (9a in Figure 2) could be regarded as being derived from a cage-like As 8 structure by attaching an As atom at one side. The isomers (C 2v ) with a higher symmetry (9b,c in Figure 2) were less stable based on our VASP calculations. It also hinted to us that the ground structure of As 8 might be a stable cluster with a magic number. From further calculations, we found that the cage-like As 8 structures unit served as the primary building unit for forming the As clusters with larger sizes. structure is also predicted in Refs. [19][20][21]. Its energy was lower than the structure with Cs symmetry (7b in Figure 1) by 0.008 eV/atom and the structure with Cs symmetry (7c in Figure 1) by 0.053 eV/atom. The wedge-like structure that looks like a cage with C2v symmetry was obtained as the lowest energy structure for As8, as seen from 8a in Figure 1. It was energetically lower than the structure with C2v symmetry (8b in Figure 1) by 0.038 eV/atom and the structure with Cs symmetry (8c in Figure  1) by 0.048 eV/atom. We also checked the cage structure with Oh symmetry cut from the bulk phase reported by Baruah [23], and we found that it was 0.64 eV energy higher than our ground state structure.
After analysis of the ground structures of Asn (n = 2-8) clusters, we find that As2 was a onedimensional bridge, As3 was a two-dimensional isosceles triangle and As4 became a threedimensional tetrahedron. When n was larger than 3, the two-dimensional cluster structure, such as 4b and 5c in Figure 1, sorts more and more backward energetically. We can conclude that the structure of small As clusters tends towards a three-dimensional cage structure and it was not stable for a 2-D planar structure.

Asn (n = 9-18) Clusters
The ground-state structure of As9 with Cs symmetry (9a in Figure 2) could be regarded as being derived from a cage-like As8 structure by attaching an As atom at one side. The isomers (C2v) with a higher symmetry (9b,c in Figure 2) were less stable based on our VASP calculations. It also hinted to us that the ground structure of As8 might be a stable cluster with a magic number. From further calculations, we found that the cage-like As8 structures unit served as the primary building unit for forming the As clusters with larger sizes.  The lowest-energy structure (C 2v ) of As 10 consistd of the cage structure of 8a in Figure 1 that was edge-capped by each As atom. After relaxation, the upper bond broke to form two four-atom cages and an As 2 bridge. The structure 10b in Figure 2 was a new structure discovered by our GA global searching. The structure obtained by Zhao [22] is structure 10c in Figure 2, whose energy was higher than 10b in Figure 2 by 0.006 eV/atom and 10a in Figure 2 by 0.009 eV/atom.
The ground structure of As 11 was 11a in Figure 2 which was formed on the base of As 10 with an As atom added above the As 2 dimer and linked with an As 8 cage. It was more stable than the 11b in Figure 2 isomer with C s symmetry by 0.008 eV/atom and 11c in Figure 2 isomer by 0.04 eV/atom with C s symmetry in energy.
For As 12 , the structure with D 3d symmetry was confirmed to be the lowest energy structure among all the structural candidates considered. The highly-symmetric structure was shaped of two As 8 cages that share a four-atom plane. From another point of view, the ground-state structure of As 12 was a layered structure of three layers of atoms (3 + 6 + 3). Such a nice structure was energetically lower than 12b in Figure 2 with C 1 symmetry by 0.023 eV/atom and 12c in Figure 2 with C s symmetry by 0.025 eV/atom.
Viewing the ground-state structure of As 13 carefully, we also found two As 8 cages. Different from As 12 , the two cages jointly owned a three-atom plane. It had 0.008 eV/atom less energy than the structural 13b in Figure 2 (C 1 ) and 0.009 eV/atom less than the structure 13c in Figure 2 (C s ).
As shown in the picture, As 14 with Cs symmetry could be considered to be composed of an As atom link to the three-atom plane on one side of the As 13 (13a in Figure 2). The less stable isomer 14b was a distorted structure of 14a, which was energetically higher by 0.011eV/atom. The 14c was a structure without an As 2 dimer bridge, it was linked by As 8 cage with an As 6 cage and had 0.032 eV/atom more energy than 14a in Figure 2.
The ground-state structure of As 15 with C s symmetry was composed of two connected cages (As 8 and As 7 ). The other two candidates 15b in Figure 2 with C s symmetry and 15c in Figure 2 with C s symmetry were less stable than the ground-state structure by 0.005 eV/atom and 0.025 eV/atom in energy, respectively.
An upward As 8 cage and a downward As 8 cage connected to form a new structure as the ground state of As 16 (16a in Figure 2 with C 2h symmetry). It was more stable than the C 2 -symmetry isomer 16b in Figure 2 and the C s -symmetry isomer 16c in Figure 2 by 0.05 eV and 0.08 eV in energy. Although structural 16c in Figure 2 contained an As 8 cage and an As 2 bridge, the As 6 cage in the structure led to the overall energy as being higher than other two isomers.
The lowest energy structure of As 17 (17a in Figure 2) with C s symmetry was built by As 8 and As 7 units with an As 2 bridge in the middle. The two C s -symmetry isomorphic structure 17b,c in Figure 2 were also formed by the As 8 and As 7 units with different orientation connections.
The ground-state of As 18 (18a in Figure 2) with C 2v symmetry was formed by two identical As 8 units and an As 2 bridge in the middle. Our structure was exactly the same as that in Ref. [22]. Two slightly higher energy isomers (18b in Figure 2) with C 2h symmetry and 18c in Figure 2 with C 2v symmetry were combined by the same units as 18a in Figure 2 with different orientations, and they were energetically higher than 18a in Figure 2 by 0.05 eV and 0.35 eV energy. The calculations showed that the structure with As 8 unit and As 2 bridge in the middle was more stable than other cage structures. We also generated one As 18 structure cut from bulk phase and the energy was 3.17 eV higher than the ground state. So, we think the chain structure with As 8 units and an As 2 bridge is much more important for middle-sized As n clusters.

As n (n = 19-24) Clusters
The ground-state structure of As 19 (19a in Figure 3) with C s symmetry could be regarded as an As atom added into one side of As 18 . Therefore, As 19 (19a in Figure 3) could be considered as the combination of As 8 -As 2 -As 8 -As 1 . The isomers of As 19 (19b,c in Figure 3) with only C 1 symmetry were both built up by two identical As 8 units and an As 3 bridge. Due to the distortion of the structures, they had 0.008 eV/atom and 0.009 eV/atom higher energy than 19a in Figure 3. The ground-state structures of As20 was predicted to be C1 symmetry, as shown in 20a in Figure  3. It could be regarded as an As8 cage link with an As10 cage joined by an As2 bridge. As20 (20a in Figure 3) could be considered as the combination of As8-As2-As10. Zhao [22] predicted the optimal combinations for the As20 is super-clusters of As4-As2-As8-As2-As4. The energy of As20 (20a in Figure  3) we got based on the genetic algorithm was energetically lower than the structure 20b in Figure 3 with C2v symmetry. A distorted structure 20c in Figure 3 originated from 20a in Figure 3 also appeared in our calculations. After the calculations with VASP, two isomers showed 0.006 eV/atom and 0.053 eV/atom higher energy than 20a in Figure 3. This result shows the super-clusters of As4-As2-As8-As2-As4 did not have much of an advantage.
For As21, the most stable structure (21a in Figure 3) could be regarded as an As atom link to the As10 cage of As20 (20a in Figure 3). Two other isomers (21b,c in Figure 3) were constituted by an As8 cage and an irregular As11 structure linked with an As2 bridge. They were energetically higher than 21a in Figure 3 by 0.011 eV/atom and 0.017 eV/atom.
Rather than simply increasing the number of atoms on the edge, the ground-state structure of As22 with Cs symmetry are formed with two symmetrical As10 cages in the As20 units and an As2 bridge in the middle. The isomers were two kinds of super-clusters (22b in Figure 3, As6-As2-As8-As2-As4 and 22c in Figure 3, As4-As2-As8-As8), and they were stretched by more units compared to the groundstate structures (22a in Figure 3). Although we could find stable As8 and As4 units in isomers, they each had 0.02 eV and 0.09 eV higher energy than 22a in Figure 3. According to previous findings, it can be found that the second lower energy As10 cage 10b in Figure 2 will be favorable if the structure is composed by an As8 unit connected with an As2 bridge.
The ground-state structures of As23 (23a in Figure 3) with C2v symmetry seemed to be four As8 cages linked to each other to share the three-atom plane, and the bottom edge of the middle two cages were broken. At the same time, we could also regard 23a in Figure 3 as a super-cluster of As8-As2-As3-As2-As8. The other two candidates, 23b in Figure 3 with Cs symmetry and 23c in Figure 3 with C1 symmetry, were less stable than the structure 23a in Figure 3 by 0.008 eV/atom and 0.021 eV/atom in energy, respectively.
The lowest energy structure of As24 with Cs symmetry (24a in Figure 3) was built by three units (an As4 cage and two identical As8 cages), connecting the neighboring structure with an As2 bridge. Zhao [22] considered that the optimal combinations of the super-clusters As24 are As6-As2-As8-As2-As6 (24b in Figure 3) with C2v symmetry. DFT calculations show that the ground-state structure of As24 we get based on the genetic algorithm weare the combinations of As8-As2-As8-As2-As4, which was energetically lower than the structure 24b in Figure 3 by 0.018 eV/atom. Besides, we also gained The ground-state structures of As 20 was predicted to be C 1 symmetry, as shown in 20a in Figure 3. It could be regarded as an As 8 cage link with an As 10 cage joined by an As 2 bridge. As 20 (20a in Figure 3) could be considered as the combination of As 8 -As 2 -As 10 . Zhao [22] predicted the optimal combinations for the As 20 is super-clusters of As 4 -As 2 -As 8 -As 2 -As 4 . The energy of As 20 (20a in Figure 3) we got based on the genetic algorithm was energetically lower than the structure 20b in Figure 3 with C 2v symmetry. A distorted structure 20c in Figure 3 originated from 20a in Figure 3 also appeared in our calculations. After the calculations with VASP, two isomers showed 0.006 eV/atom and 0.053 eV/atom higher energy than 20a in Figure 3. This result shows the super-clusters of As 4 -As 2 -As 8 -As 2 -As 4 did not have much of an advantage.
For As 21 , the most stable structure (21a in Figure 3) could be regarded as an As atom link to the As 10 cage of As 20 (20a in Figure 3). Two other isomers (21b,c in Figure 3) were constituted by an As 8 cage and an irregular As 11 structure linked with an As 2 bridge. They were energetically higher than 21a in Figure 3 by 0.011 eV/atom and 0.017 eV/atom.
Rather than simply increasing the number of atoms on the edge, the ground-state structure of As 22 with C s symmetry are formed with two symmetrical As 10 cages in the As 20 units and an As 2 bridge in the middle. The isomers were two kinds of super-clusters (22b in Figure 3, As 6 -As 2 -As 8 -As 2 -As 4 and 22c in Figure 3, As 4 -As 2 -As 8 -As 8 ), and they were stretched by more units compared to the ground-state structures (22a in Figure 3). Although we could find stable As 8 and As 4 units in isomers, they each had 0.02 eV and 0.09 eV higher energy than 22a in Figure 3. According to previous findings, it can be found that the second lower energy As 10 cage 10b in Figure 2 will be favorable if the structure is composed by an As 8 unit connected with an As 2 bridge.
The ground-state structures of As 23 (23a in Figure 3) with C 2v symmetry seemed to be four As 8 cages linked to each other to share the three-atom plane, and the bottom edge of the middle two cages were broken. At the same time, we could also regard 23a in Figure 3 as a super-cluster of As 8 -As 2 -As 3 -As 2 -As 8 . The other two candidates, 23b in Figure 3 with C s symmetry and 23c in Figure 3 with C 1 symmetry, were less stable than the structure 23a in Figure 3 by 0.008 eV/atom and 0.021 eV/atom in energy, respectively. The lowest energy structure of As 24 with C s symmetry (24a in Figure 3) was built by three units (an As 4 cage and two identical As 8 cages), connecting the neighboring structure with an As 2 bridge. Zhao [22] considered that the optimal combinations of the super-clusters As 24 are As 6 -As 2 -As 8 -As 2 -As 6 (24b in Figure 3) with C 2v symmetry. DFT calculations show that the ground-state structure of As 24 we get based on the genetic algorithm weare the combinations of As 8 -As 2 -As 8 -As 2 -As 4 , which was energetically lower than the structure 24b in Figure 3 by 0.018 eV/atom. Besides, we also gained another high symmetric structure with C 2v symmetry (24c in Figure 3) that was constituted by three As 8 units. However, its energy was 0.025 eV/atom higher than the ground state structure (24a in Figure 3). We could realize from this result that the lowest energy structures of larger As clusters not only have the combination of As 8 units but also needed an As 2 bridge in the middle of adjacent units.

As n (n = 2-24) Charged Clusters
We also studied the ground structures of As n (n = 2-24) charged clusters. Theoretically, it was easy to simulate a cationic or anionic cluster by adjusting the total electrons from the neutral cluster. From Figures S2 and S3, we could know the lowest energy structures of charged clusters (n = 2-4) were almost the same as with neutral cases. For As 8 clusters, 8a in Figure 1 structures were quite stable even in cationic or anionic cases and it was the cluster with the magic number. For other As n (n < 16) clusters, the isomers changed the energy sequence as the system changed the electron numbers. It was interesting to find that the structures for As n (15 < n < 24) clusters were quite stable whatever attachment of extra electron to the neutral or losing of an electron from the neutral cluster.

As n (n = 28, 38, 40, 180) Clusters
With the increase of cluster size, it was more and more difficult to exhaust all possible local minimum structures. We tried to study the larger clusters As 28 , As 38 , and As 40 based on the above findings. The structural size evolution and electronic properties of arsenic clusters indicated that the clusters combined by an As 2 bridge and an As 8 cage had lower energy than their isomers and showed more stability in each local size-dependent range. Here we have to emphasize that As 4 and As 6 units were not dominant for the larger As n cluster, which was different from Zhao's result [22]. Furthermore, different sizes of fullerene cage structure isomers were also calculated to compare with our ground state structures in energy, and their energies were far more than units-linked one-dimensional structures. Given this understanding, we constructed As 28 as As 8 -As 2 -As 8 -As 2 -As 8 and As 38 as As 8 -As 2 -As 8 -As 2 -As 8 -As 2 -As 8 in all possible ways. The structures we calculated of As 28 are listed along with the increase of energy in Figure 4. The lowest energy structure of As 28 with C 2v symmetry ( Figure 4a) had 0.05 eV less energy than the structure with C s symmetry ( Figure 4b) and 0.10 eV less than the structure with C 2v symmetry (Figure 4c). In addition to considering structure growth in the one-dimensional direction, we also calculated the longitudinal growth mode. Three isomers Figure 4c,e,f were all with C s symmetry and energetically higher than the lowest energy structure a with 0.08 eV, 0.35 eV, and 0.40 eV, respectively. Besides, the other two semi-ring isomers Figure 4g,h had 0.44 eV and 0.70 eV energy higher than the ground state structure. Compared with the bulk truncated structure of As 28 , the lowest energy structure of As 28 was 0.16 eV/atom lower.
The lowest energy structure of As 38 with C 2v symmetry ( Figure 5a) and its isomers are listed in Figure 5. We could find that the structure of Figure 5a was a continuation of As 8 , As 18 , and As 28 clusters, and all of them were constructed with As 8 cages in the same direction with As 2 bridges. The structural isomers Figure 5b with C 2v symmetry and Figure 5c with C 2h symmetry could be regarded as one-dimensional chain structures, the same as Figure 5a. Their respective energies were 0.10 eV and 0.22 eV higher than Figure 5a. The C 2v isomer Figure 5d could be regarded as a two-dimensional structure that has four As 8 cages held in four directions and all of them point to the center. The energy of this structure was highest in our calculation with 1.11 eV higher energy than a. Compared with the bulk truncated structure of As 38 , the lowest energy structure of As 38 was 0.156 eV/atom lower. We could find that the structures did not change previous growth tendencies even with increasing the size of the clusters.  Considering that As40 can form ring and fullerene cage structures, we also studied the structures of As40. The structures we calculated of As40 are listed along with the increase of energy in Figure 6. We found that the lowest energy structure of As40 with C1 symmetry (Figure 6a) and its isomers Figure 6b (0.42 eV energy higher) with Cs symmetry both were one-dimensional chain structures. Three isomers Figure 6c,d,e could all be regarded as two-dimensional structures and energetically   Considering that As40 can form ring and fullerene cage structures, we also studied the structures of As40. The structures we calculated of As40 are listed along with the increase of energy in Figure 6. We found that the lowest energy structure of As40 with C1 symmetry (Figure 6a) and its isomers Figure 6b (0.42 eV energy higher) with Cs symmetry both were one-dimensional chain structures. Three isomers Figure 6c,d,e could all be regarded as two-dimensional structures and energetically Considering that As 40 can form ring and fullerene cage structures, we also studied the structures of As 40 . The structures we calculated of As 40 are listed along with the increase of energy in Figure 6. We found that the lowest energy structure of As 40 with C 1 symmetry (Figure 6a) and its isomers Figure 6b (0.42 eV energy higher) with C s symmetry both were one-dimensional chain structures. Three isomers Figure 6c,d,e could all be regarded as two-dimensional structures and energetically higher than the lowest energy structure a with 0.014 eV/atom, 0.044 eV/atom, and 0.09 eV/atom, respectively. Seemingly stable three-dimensional fullerene cage isomers Figure 6f with D 5d symmetry and Figure 6g with D 5d symmetry were 0.194 eV/atom and 0.226 eV/atom higher in energy than the lowest energy structure.
Materials 2018, 11, x FOR PEER REVIEW 9 of 13 higher than the lowest energy structure a with 0.014 eV/atom, 0.044 eV/atom, and 0.09 eV/atom, respectively. Seemingly stable three-dimensional fullerene cage isomers Figure 6f with D5d symmetry and Figure 6g with D5d symmetry were 0.194 eV/atom and 0.226 eV/atom higher in energy than the lowest energy structure. Based on the finding above, we could construct the ring structure of an As180 cluster based on the As8 units and As2 bridge, which is shown in Figure 7. The HOMO-LUMO gap of As180 was 1.868 eV and the binding energy per atom was −2.901 eV.  Based on the finding above, we could construct the ring structure of an As 180 cluster based on the As 8 units and As 2 bridge, which is shown in Figure 7. The HOMO-LUMO gap of As 180 was 1.868 eV and the binding energy per atom was −2.901 eV.
Materials 2018, 11, x FOR PEER REVIEW 9 of 13 higher than the lowest energy structure a with 0.014 eV/atom, 0.044 eV/atom, and 0.09 eV/atom, respectively. Seemingly stable three-dimensional fullerene cage isomers Figure 6f with D5d symmetry and Figure 6g with D5d symmetry were 0.194 eV/atom and 0.226 eV/atom higher in energy than the lowest energy structure. Based on the finding above, we could construct the ring structure of an As180 cluster based on the As8 units and As2 bridge, which is shown in Figure 7. The HOMO-LUMO gap of As180 was 1.868 eV and the binding energy per atom was −2.901 eV.

Electronic Properties of As n Clusters
The binding energy per atom for the ground states of As n (n = 2-24) clusters are shown in Figure 8a. In the size range of n = 12-24, the binding energy increased smoothly with weak odd-even oscillation properties. This result can be related to the evolution of the ground-state from cage-like structure to cage-link structure at n = 12. Besides, the binding energy of As 8 was a peak value in the small size of n = 3-11, and this suggests the ground structure of As 8 would be a vital growth unit in larger structures. Our next calculation also proved the conjecture.

Electronic Properties of Asn Clusters
The binding energy per atom for the ground states of Asn (n = 2-24) clusters are shown in Figure  8a. In the size range of n = 12-24, the binding energy increased smoothly with weak odd-even oscillation properties. This result can be related to the evolution of the ground-state from cage-like structure to cage-link structure at n = 12. Besides, the binding energy of As8 was a peak value in the small size of n = 3-11, and this suggests the ground structure of As8 would be a vital growth unit in larger structures. Our next calculation also proved the conjecture.  In Figure 8b, we present the energy gaps between the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) for the lowest-energy state of As n (n = 2-24) clusters. As is known, the cluster with the larger energy gap is more stable and easier to prepare. As the largest energy gap of As 4 is 4.06 eV, it is the most prominent species in arsenic vapor, leading to a number of experimental and theoretical studies on As 4 clusters. Although the gaps of As n clusters from n = 5-10 change smoothly, we also observe the gap of As 8 is highest locally, which points out that the stability of As 8 was higher than the neighboring cluster. The HOMO-LUMO gap was higher for As n (n = 4, 6, 8, 12, 14, 16, 18, 20, 22, and 24) than their adjacent structures. The atoms in these even-numbered sequences were all three-coordination and eight-electron structure. The odd-numbered clusters were unable to achieve this condition, so they were less stable than the odd-numbered species.
In clusters physics, the second-order difference of cluster energy is a more sensitive datum to reflect the stability of clusters. We plotted the second-order difference of cluster energies defined by ∆ 2 E = E(n + 1) + E(n − 1) − 2E(n). Figure 8c describes how the second-order differential energy changed with the increase of atom number and it shows good odd-even oscillation properties. The second-order difference of cluster energies of even-numbered clusters were all higher than their adjacent odd-numbered clusters. Therefore, we could draw the conclusion that even-numbered clusters were more stable than their neighboring odd-numbered clusters. Above all peaks for ∆ 2 E, three local maximum peaks were found at n = 4, 8, and 18, where As n (n = 4, 8, and 18) clusters were chemically stable.

Conclusions
We have adopted the genetic algorithm and all-electron DFT calculations to systematically study the structures and electronic properties of As n (n = 2-24). The ground-state structures of As n clusters change from two to three dimensional after n = 3. Arsenic clusters followed a structural growth pattern starting from n = 8 and the structures of As n (n = 9-24) clusters could all be regarded as evolving from the ground state structure of an As 8 unit and an As 2 bridge. The binding energy of As n (n = 2-24) clusters had a periodic step-like behavior, and the size-dependent HOMO-LUMO gap and second-order difference of cluster energies exhibited obvious even-odd alternations with several magic numbers. Based on the growth pattern concluded from small As n clusters, the possible superstructures of As 28 , As 38, As 40 , and As 180 were discussed.