Stability and bonding nature for icosahedral or planar cluster of hydrogenated boron or aluminum

Ab initio molecular orbital calculations are performed for B 13 − , Al 13 − , B 12 H 122 − , Al 12 H 122 − , Si 10 , and Si 10 H 16 clusters. The highest occupied molecular orbital (HOMO) of stable and unstable clusters is bonding and antibonding orbitals, respectively. The cluster size dependences of the orbital energies are almost the same for B 13 − and Al 13 − icosahedral clusters, when the size and the energy are properly normalized. The normalized factors for size and energy are almost coincident with the ratios of those of the atomic outer s orbitals. On the other hand, the most stable size of B 13 − is smaller than that of Al 13 − , and this ratio of the stable size seems to be affected by the ratio of the sizes of the atomic outer p orbitals. As a result, B 13 − and Al 13 − icosahedral clusters have antibonding and bonding orbitals for HOMOs and so are unstable and stable, respectively. The situation for B 13 − and Al 13 − planar clusters is opposite to that discussed above for the icosahedral clusters. The orbital energies for the metallic bonding Al 13 − icosahedral and Si 10 clusters can be reproduced by the Woods–Saxon model; however, those for the unstable B 13 − icosahedral and covalent bonding B 12 H 122 − , Al 12 H 122 − , and Si 10 H 16 clusters cannot be reproduced by the Woods–Saxon model. After optimization of the parameters of the Woods–Saxon model for the Al 13 − icosahedral and Si 10 clusters, the orbital energies are reproduced very well and the sizes and shapes of the potential are reasonable.


I. INTRODUCTION
A cluster is a group of a countable number of atoms. The cluster has an interesting nature and a structure that is different from the bulk solid or an isolated atom or molecule because of the special cluster size. Many computational studies have been performed to predict the properties of clusters, and it has been proposed that films with new functions may be obtained by controlling the structure of clusters. A Si cluster with a transition metal atom 1 and a boron hydride cluster as a boron dopant 2 are examples of structurally controlled clusters. The boron hydride cluster also has other uses because it changes its stable structure depending on the degree of hydrogenation. A boron hydride cluster BnHm has an icosahedral structure, as observed in a boron crystal, but a boron cluster Bn has a planar structure. [3][4][5] The planar structure is expected to construct a plane with a graphenelike structure that has a hexagonal and triangular lattice. [6][7][8] The aluminum hydride crystal (AlH 3 ) is promising as a hydrogen storage material because it can contain up ARTICLE scitation.org/journal/adv to 10.1 wt. % of H atoms. However, Al crystals hardly react with H 2 at normal temperature and pressure. 9 This problem can be solved by using an Al cluster because the energy barrier of the physical hydrogen adsorption reaction of an Al cluster is smaller than that of an Al crystal. 10,11 Investigations using density functional theory (DFT) calculations have revealed that more Al or B atoms contained in the cluster will result in a lower energy barrier. 12 BnHm releases molecular B 2 H 6 when it adsorbs more hydrogen than the amount expressed by the Wade law. The cluster, as a building block, is useful to understand the properties of a crystal. Quasicrystals and their approximants are examples of cluster solids, for example, one containing an icosahedral Al 12 cluster. In such a solid, the Al 12 cluster converts its bonding nature without changing the icosahedral structure. 13,14 Al 12 clusters with a rhenium (Re) atom at their center have a metallic bond and those without a Re atom have a covalent bond. This bonding conversion changes the electrical property of the whole bulk solid. B is a group-13 element, the same as Al, and there exists an icosahedral B 12 cluster in a boron crystal. Different from Al, other atoms are not located at the center of the cluster, but the B crystal changes the covalent bond to a metallic bond by the vanadium atom being doped in another site. 15,16 All of these bonding conversions are caused without collapsing the icosahedral cluster structure. Isolated clusters change the metallic bond to a covalent bond by hydrogenation. A Si 10 cluster has a metallic bond, and its structure is compact, but hydrogenated Si 10 H 16 has a covalent bond, and it forms a diamondlike structure. 17,18 Although Al and B belong to the same group in the periodic table, the bonding nature and stable structure of isolated Al and B clusters are different. An Al cluster changes the metallic bond to a covalent bond by hydrogenation with the same icosahedral structure [Figs. 1(a)-1(c)]. 19 On the other hand, the B cluster forms a planar structure [ Fig. 1(b)], and the structure is changed from planar to icosahedral by hydrogenation [ Fig. 1(c)]. 5 It is proposed that the bonding conversion is a π bond to σ bond. Typically, it is known that a cluster with a specific number of atoms or valence electrons is more stable than other clusters, and this special number is called the magic number. The magic number depends on the geometric shell model like a rare gas cluster, 20 the electronic shell model like an alkali metal cluster, 21 or the stability of molecular bonding like a fullerene. 22 For a metallic bond cluster, the Woods-Saxon model can be applied. 23 The Woods-Saxon model is a kind of free electron approximation. In this model, the cluster shape is approximated to a sphere and the valence electrons are influenced by the interface of the cluster. As shown in Fig. 2, as an intermediate model between a harmonic oscillator and a square well, the potential for the electron can be expressed as 21,23 FIG. 1. Stable structure of B or Al clusters.

FIG. 2.
Schematic drawing of eigenvalues of a harmonic oscillator, Woods-Saxon model, and square well potential.
Here, r is the distance from the center of the cluster, R is the radius of the cluster, V is the depth of the potential, and a is the diffuse parameter which corresponds to smoothness at the boundary of the cluster. The magic numbers of valence electrons calculated using this model are 2, 8, 18, 20, 34, 40, 58, . . .. When a cluster follows this model, the orbital energy levels can be classified as 1s, 1p, 1d, 2s, 2p, 1f, . . ., by the orbital structure and degeneracy, like an atom. If the number of valence electrons in the cluster is one of the magic numbers, the cluster becomes closed shell and electronically stable. For example, a Nan cluster is a typical metallic bonding cluster, and it follows this model well. 21 Hence, in a metallic bonding cluster, valence electrons can be approximated as free electrons. In other words, when the orbital level follows this model well, the bonding condition of the cluster can be judged to be a metallic bond. An Al 13 − cluster has 40 valence electrons, and 40 is the magic number for occupying up to a 2p orbital, so it is very stable. 24 Al 13 − is also geometrically stable because it forms an icosahedral structure with a center atom [ Fig. 1(a)]. According to these two reasons, an Al 13 − cluster is more chemically stable than other Al clusters, and it hardly reacts with oxygen, whereas the others do react with oxygen. 25 However, B 13 − forms a planar structure [ Fig. 1(b)] and when in an icosahedral structure [ Fig. 1(a)], it is unstable, not even with a locally stable structure.
In this paper, we investigate B 13 − , Al 13 − , B 12 H 12 2− , Al 12 H 12 2− , Si 10 , and Si 10 H 16 clusters. To clarify the contribution of the cluster orbital to the structural stability, we investigated the orbital energies as a function of the radius of the clusters. We also examined the dependence of the structural stability on whether the highest occupied energy orbital (HOMO) is bonding or antibonding. Finally, we compared the energy levels of the clusters with those of the ARTICLE scitation.org/journal/adv Woods-Saxon model to judge if the cluster has metallic bonding or not.

II. CALCULATION METHOD
We calculated the cluster structure and orbital energy using Gaussian 03W and Gaussian 09W based on density functional theory (DFT). 26,27 To investigate whether the B3LYP functional can be used for the hydrogenated Al cluster, we calculated the bonding energy and wavenumber of the Al-H molecule using 9 functionals at the 6-31g(d) level and compared the results of the calculation and experiments. [28][29][30][31][32] We applied the B3LYP/6-31g(d) level to a boron hydride cluster and succeeded in explaining the experimental results. 5 13 − structures. Next, the bond lengths were optimized by fixing the bonding angles when the total energy was minimized.
We also calculated the cluster radius dependence of the orbital energies by fixing the bond angle. Here, the Al 13 − , B 13 − , and Si 10 clusters have 40 electrons, which is one of the magic numbers for a metallic bonding cluster. The hydrogenated clusters Al 12 H 12 2− and B 12 H 12 2− have 50 electrons, and Si 10 H 16 has 56. These are not magic numbers.

III. DETERMINATION OF A SUITABLE FUNCTIONAL FOR THE CLUSTERS
It was determined that the most suitable functional for the optimization of a boron hydride cluster was B3LYP. 5,33 Experimentally, the binding energy of the Al-H bond, 28,29 infrared absorption spectra, 28  be applied for the aluminum hydride cluster, we calculated the AlH molecule using the generalized gradient approximation (GGA) functionals BLYP, PW91, BPBE, PBE, TPSS, B3LYP, B3PW91, PBE0, and TPSSh at the 6-31g(d,p) level. We compared the calculated results and experimental values of the binding energy and the vibrational wavenumber of Al-H. The value of the binding energy of Al-H calculated with B3LYP was within the range of the error of the experimental values (Fig. 3). The value of the calculated vibrational wavenumber slightly deviated from the experimental range (Fig. 4). The structure of Aln (n = 1-13) was optimized in each functional, and the vertical ionization energy (vIE) (eV) was calculated as the energy difference between Aln and Aln + , which was then compared with that determined by experiment. The value for each n is shown in Fig. 5, and the average of the absolute error for the B3LYP functional was the lowest. Therefore, by considering the reproducibility of the Al-H bond and Aln clusters, it was confirmed that B3LYP can be used for aluminum hydride clusters. A similar accuracy can be expected for Si clusters.  Fig. 6(a)]. Orbital f is degenerated in 7 and g in 9, but they are resolved in 1fa and 1fb, and in 1ga and 1gb, respectively, because both ICOSAHEDRON and CAGE clusters have icosahedral symmetry As for all clusters, the cluster size when the total energy is minimized is called the most stable size. The sizes refer to the distances between atoms other than H and the cluster center of ICOSA-HEDRON and CAGE, or an average of nearest neighbor atomic distances other than those to H of the other clusters. Lengths of B-H, Al-H, and Si-H were fixed at those of the most stable size cluster. The relationships between the cluster size and orbital energy are shown in Fig. 7. Most of the occupied orbitals (solid lines) were the bonding orbitals, the energies of which decreased with decreasing the cluster size. On the other hand, most of the unoccupied orbitals (broken lines) were the antibonding orbitals.  most stable sizes of the former and the latter are located at the left and the right of the cross point of the 1fa and 2s orbitals, respectively. However, although the size dependence of the energy levels is similar, the relative positions of the most stable size are different.
The case of the silicon cluster is shown in Fig. 7(c). The HOMOs of both stable Si 10 and Si 10 H 16 are bonding orbitals. For all cases investigated in this work, the HOMO of stable clusters was bonding, but the HOMO of unstable clusters was antibonding around the most stable size. Therefore, the most stable structure of each cluster seems to be determined by whether the HOMO is bonding or antibonding.
Focusing on B and Al clusters with the same structure, the relationships between the cluster size and orbital energy exhibited a similar tendency, but the positions of the most stable size were smaller and larger for B and Al, respectively. As for Al 13 − (ICOSA-HEDRON) and B 13 − (ICOSAHEDRON), the size dependences of the orbital energy of each cluster showed a similar tendency, but the most stable size of Al 13 − (ICOSAHEDRON) was located at the longer side of the cross point and that of B 13 − (ICOSAHEDRON) at the shorter side. It seems that this difference originated from the size difference of the B and Al atomic orbitals. We normalized the two axes of B 13 − (ICOSAHEDRON) and Al 13 − (ICOSAHEDRON), that is, the energy and size axes for B were reduced and expanded, respectively, as the cross point of the 1fb and 2p orbitals and each line overlapped as much as possible (Fig. 8). Table I  energies and the radii of B and Al (italic numbers in Table I), respectively. The ratio of the cluster orbital energy to radius seemed to be strongly dependent on the atomic s orbitals. On the other hand, the ratio of the cluster radii of the most stable size was sufficiently larger than the above value, and it was considered to be influenced by the ratio of the sizes of the p atomic orbitals (bold numbers in Table I).
Although the radii of the atomic orbitals of 2s and 2p for B were near, the radius of 3p was larger than that of 3s for Al because the 2p

V. VALIDITY OF THE WOODS-SAXON MODEL FOR METALLIC BOND OF CLUSTER
The words "metal" and "metallic bond" are often confused, so we first define these in Fig. 9. 14 Metal elements become stable ions by releasing a few electrons. These electrons (valence electrons) spread through the whole system, and the system gets cohesive energy because the kinetic energy of the electrons decreases. These ions and electrons condense owing to the Coulomb attraction. This is the condensing mechanism for metal solids. Metal atoms in metallic bonding clusters can be estimated to bond with the same mechanism. In a metallic bonding solid, the valence electron behaves as a free electron and it becomes a metal. A metal is a solid whose density of state [D(ε)] at the Fermi energy (ε F ) is finite and the wave function at ε F is extended; therefore, the electrical conductivity is not zero even at a temperature of absolute zero. On the other hand, when D(ε F ) is zero or the wave function at ε F is localized, the solid becomes nonmetallic, that is, an insulator or semiconductor. Focusing on a cluster, it has very few atoms and the energy level is not continuous. Accordingly, a cluster cannot be a metal even when the HOMO is not perfectly occupied. However, the bonding of a cluster can be metallic when the valence electrons behave as free electrons in the empty sphere space, which has the same size as the cluster, that is, the valence electron system can be described by the Woods-Saxon model. In short, energy levels of cluster orbitals correspond to those of the model when the cluster has a metallic bond.
We calculated the orbital energy level of each cluster and compared it with the eigen energy level of the Woods-Saxon model [ Figs. 10(a) and 10(b)]. The lines on the left side refer to the energy levels of the Woods-Saxon model, and the lines on the right side refer to the calculated orbital energy levels of each cluster. The left side is normalized as the heights of the 1s and 2p lines become the same as those on the right side. It is possible to compare only the order of the orbital energies.
Al 13 − (ICOSAHEDRON) and Si 10 , which have a metallic bond follow the Woods-Saxon model. The HOMO of Al 13 − (ICOSA-HEDRON) is a 1fb orbital, and its energy is higher than that of the 2p orbital, but the order of the orbital energy is not interchanged because these two orbitals have almost the same energy. The 1d and 1f orbitals of Si 10 are divided into many levels because

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Si 10 has lower symmetry, but the order of the orbital group is not interchanged. These results are consistent with the fact that Al 13 − (ICOSAHEDRON) and the Si 10 clusters have a metallic bond. The other clusters did not follow the Woods-Saxon model. As for B 13 − (ICOSAHEDRON), the 1fa energy is higher than 2p and the levels are interchanged. This resulted from this cluster being an unstable cluster. Al 12 H 12 2− (CAGE), B 12 H 12 2− (CAGE), and Si 10 H 16 do not follow the Woods-Saxon model because they have a covalent bond. As for Al 12 H 12 2− (CAGE), the 1fa level is lower than 2s and 1fb is higher than 1ga. As for B 12 H 12 2− (CAGE), 1fb is higher than 1ga. 1d and 1f of Si 10 H 16 resolve into 2 and 3 levels because of low symmetry. Then 1d, 1f, and 1g are interchanged complicatedly. The model cannot be used to compare with the B 13 − (PLANE) and Al 13 − (PLANE) orbital energies because they are far from spherical.
As a result, only Al 13 − (ICOSAHEDRON) and Si 10 followed the Woods-Saxon model because these clusters have a metallic bond. Orbital shape and degeneracy depended on the symmetry of the clusters, and the order of the orbital level reflects the bonding nature of the clusters.
The potential, V(r) of the Woods-Saxon model, is given as Eq. (1). To find parameters suitable for orbitals of the Al 13 − and Si 10 metallic bonded clusters, we solved the Schrödinger equation in the radial direction using the difference method.
By considering metallic bonded clusters as having spherical symmetry, their potential can be approximated by the Woods-Saxon potential. Schrödinger's equation can be solved numerically by transforming it into a difference equation, which is a kind of diagonalization problem. We made a computer program to find the

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scitation.org/journal/adv eigenvalues at any V(r) parameters. We determined the most suitable parameters V, R, and a by comparing the solutions E calc and Gaussian results E Gaussian for each cluster. The solving procedure for the Al 13 − and Si 10 clusters was composed of three steps. (1) Coordinate the 1s and 2p eigenvalues to that of the result of Gaussian by fitting V. (2) Calculate the root mean square of the difference of each orbital energy (Σ(E Gaussian − E calc ) 2 ) 1/2 . A lower root mean square will result in more suitable parameters. (3) Perform the same calculation by changing R and a, and determine the most suitable parameters V, R, and a [ Fig. 11(a)]. We show the shapes of V(r) [Fig. 11(b)] and the orbital energy [ Fig. 11(c)]. The results a > R for Al 13 − and Si 10 clusters are different from a < R, which is the result for the typical metallic bonding Na cluster. 35 The shape of the potential approaches the square-well when a < R and the harmonic oscillator when a > R. Therefore, the above difference means that the bonding nature of the Na cluster is more metallic than those of Al 13 − and Si 10 clusters. On the other hand, the interchanges of energy levels as shown in unstable or covalent bonding clusters in Fig. 10 have not occurred by changing the parameters V, R, or a.
The most suitable R of Al 13 − in the Woods-Saxon model was 1.35 times the radius of the Al 13 − cluster, which is the distance between the center and the outer atom of the cluster, and R of Si 10 in the Woods-Saxon model was 1.54 times the radius of the Si 10 cluster. These results are reasonable, when the atomic radii are considered.

VI. CONCLUSION
The HOMO of a stable cluster is bonding, and the HOMO of an unstable cluster is antibonding. The overall features of cluster size dependences of the orbital energy for B 13 − (ICOSAHEDRON) and Al 13 − (ICOSAHEDRON) are almost the same, when the size and the energy are normalized. The normalized factors for size and energy are near the ratio of the radii of the atomic outer s orbitals and the ratio of the s orbital energies, respectively. On the other hand, the difference of the most stable cluster size of B 13 − (ICOSAHEDRON) and Al 13 − (ICOSAHEDRON) depends on the ratio of the atomic outer p orbitals. The 2p orbital of B does not have an inner p orbital, and the size is almost the same as the 2s orbital; however, the 3p orbital of Al has an inner 2p orbital and the size is larger than that of the 3s orbital. From these facts, the HOMOs of B 13 − (ICOSA-HEDRON) and Al 13 − (ICOSAHEDRON) with the most stable size are antibonding and unstable, and bonding and stable, respectively. When the order of the orbital energies of the cluster follows the Woods-Saxon model, the bonding nature of the cluster is metallic and a covalent bonded cluster will not follow this model. The Woods-Saxon model for Al 13 − (ICOSAHEDRON) and Si 10 with the optimized parameters provides reasonable orbital energies and potential shapes. Therefore, we propose the way to judge whether a cluster with spherical symmetry has a metallic bond or not is by investigating whether the orbital energies of the cluster follow those obtained by the Woods-Saxon model.

ACKNOWLEDGMENTS
One of the authors (H.Y.) was supported by the MERIT program and the Marubun Research Promotion Foundation. This work was supported by JSPS KAKENHI, Grant Nos. 15K14106 and