Chemical Bonding and Dynamic Structural Fluxionality of a Boron-Based Na5B7 Sandwich Cluster

Doping alkali metals into boron clusters can effectively compensate for the intrinsic electron deficiency of boron and lead to interesting boron-based binary clusters, owing to the small electronegativity of the former elements. We report on the computational design of a three-layered sandwich cluster, Na5B7, on the basis of global-minimum (GM) searches and electronic structure calculations. It is shown that the Na5B7 cluster can be described as a charge-transfer complex: [Na4]2+[B7]3−[Na]+. In this sandwich cluster, the [B7]3− core assumes a molecular wheel in shape and features in-plane hexagonal coordination. The magic 6π/6σ double aromaticity underlies the stability of the [B7]3− molecular wheel, following the (4n + 2) Hückel rule. The tetrahedral Na4 ligand in the sandwich has a [Na4]2+ charge-state, which is the simplest example of three-dimensional aromaticity, spherical aromaticity, or superatom. Its 2σ electron counting renders σ aromaticity for the ligand. Overall, the sandwich cluster has three-fold 6π/6σ/2σ aromaticity. Molecular dynamics simulation shows that the sandwich cluster is dynamically fluxional even at room temperature, with a negligible energy barrier for intramolecular twisting between the B7 wheel and the Na4 ligand. The Na5B7 cluster offers a new example for dynamic structural fluxionality in molecular systems.


Introduction
Due to the electron deficiency of boron with three valence electrons (2s 2 2p 1 ), boron and its relevant compounds and clusters have rich and unique structures, as well as unconventional chemical bonding [1][2][3][4][5][6][7][8]. Recent experimental and theoretical studies have explored the structures and bonding of a wide range of boron-based clusters. It is now known that elemental boron clusters assume two-dimensional (2D) planar or quasi-planar geometries with up to 40 atoms for anions, extended 2D sheet structures (borophenes), and three-dimensional (3D) structures such as borospherenes [8][9][10][11][12][13]. Chemical bonding in these clusters is governed by π/σ aromaticity, antiaromaticity, and conflicting aromaticity, in which electron delocalization is essential in order to compensate for the intrinsic electron deficiency of boron.

Global-Minimum and Transition-State Structures
The optimized low-lying structures of Na 5 B 7 cluster at the PBE0/6-311 + G* level are shown in the Supplementary Materials ( Figure S1). The GM Na 5 B 7 cluster is identified herein using the global searches and electronic structure calculations at three levels of theory. The Cartesian coordinates for GM C 3v ( 1 A 1 ) Na 5 B 7 cluster at the PBE0/6-311 + G* level are presented in Table S1. The PBE0/6-311 + G* method has been widely used for boron-based clusters [8,31,32]. The comparative B3LYP/6-311 + G* data for top candidate structures serve to check for the consistency of different functionals in terms of geometries and energetics. The single-point CCSD(T) calculations on the basis of optimized PBE0/6-311 + G* geometries allow benchmarking of the energetics. All three levels of theory confirm C 3v ( 1 A 1 ) structure ( Figure 1a) as the GM of Na 5 B 7 cluster. a series of boron-based sandwich clusters (B7Li4 , Na6B7 , and Na8B7 ) were lately studi theoretically, which also show dynamic fluxionality [29,30]. In 2017, Zhai and cowork discovered two virtually isoenergetic triple-layered and helix-type structures for a Be6B cluster [31]. The former sandwich structure demonstrated dual dynamic rotation/twisti modes of structural fluxionality, akin to an earth-moon system at the nanoscale.
Alkali metal elements are clearly an ideal choice for doping due to their small el tronegativities, which would allow precise tuning of the number of valence electrons i boron-based alloy cluster, one at a time. The purpose of this work is to explore how t structure of a binary Na-B cluster depends on its ratio of Na versus B components, wh underlies the stability of such a cluster structure, and whether new examples can be fered for dynamic structural fluxionality. To this end, we have reached a binary Na cluster. Computational global-minimum (GM) structure searches reveal a sandwich-ty geometry for this cluster, which features a B7 molecular wheel sandwiched in between Na4 tetrahedron and an isolated Na atom. Chemical bonding analysis suggests dou 6π/6σ aromaticity for the B7 molecular wheel, as well as 2σ aromaticity for the tetrahed Na4 ligand. In other words, the sandwich Na5B7 cluster has collectively three-fo 6π/6σ/2σ aromaticity. The sandwich shape allows the precise number of electrons to transferred from the Na4 and Na ligands to the B7 core, so that the whole sandwich syste can be stabilized via electrostatics. The cluster is faithfully formulated as [Na4] 2+ [B7] 3− [Na that is, a charge-transfer complex. The unique sandwich shape of the cluster facilita dynamic structural fluxionality, as confirmed in molecular dynamics simulations. T work also highlights the robustness of [B7] 3− as a potential inorganic ligand.

Global-Minimum and Transition-State Structures
The optimized low-lying structures of Na5B7 cluster at the PBE0/6-311 + G* level a shown in the Supplementary Materials ( Figure S1). The GM Na5B7 cluster is identifi herein using the global searches and electronic structure calculations at three levels theory. The Cartesian coordinates for GM C3v ( 1 A1) Na5B7 cluster at the PBE0/6-311 + level are presented in Table S1. The PBE0/6-311 + G* method has been widely used boron-based clusters [8,31,32]. The comparative B3LYP/6-311 + G* data for top candid structures serve to check for the consistency of different functionals in terms of geometr and energetics. The single-point CCSD(T) calculations on the basis of optimized PBE0 311 + G* geometries allow benchmarking of the energetics. All three levels of theory co firm C3v ( 1 A1) structure ( Figure 1a) as the GM of Na5B7 cluster. Among the low-lying isomers is a local-minimum (LM) C 2v ( 1 A 1 ) structure ( Figure S1), which is about 0.2 eV above the GM cluster. The remaining isomers are substantially higher in energy. Considering that adequate intramolecular charge transfers can take place from Na to B, the triplet-state structures are relatively unimportant for the present cluster system with an even number of 26 valence electrons. Indeed, our calculations show that the corresponding triplet-state structures for the top 12 singlet-state isomers in Figure S1 are 0.82-2.20 eV higher than the GM cluster at the PBE0/6-311 + G* level. Specifically, as illustrated in Figure S2, the lowest-lying triplet-state geometry of Na 5 B 7 cluster, C 1 ( 3 A), lies 0.82 eV higher above the GM structure. Other triplet-state geometries are even higher in energy, by up to 2.20 eV (relative to the singlet GM cluster), for the 12th isomeric triplet structure. Thus, the C 3v ( 1 A 1 ) cluster ( Figure 1a) is reasonably well defined on its potential energy surface as the real GM structure. Alternative structures, either singlets or triplets, are not energetically competitive, except for an LM C 2v ( 1 A 1 ) structure as mentioned above.
The top-and side-views of GM C 3v ( 1 A 1 ) Na 5 B 7 cluster are presented in Figure 1a. Basically, it is a three-layered sandwich cluster. The boron component forms a quasiplanar B 7 molecular wheel as the core layer of the sandwich. It features six-fold in-plane coordination for the central B site. On the other hand, the Na 5 component is divided into a Na 4 tetrahedron and an isolated Na atom. The two Na-based layers serve as ligands for the B 7 molecular wheel. Specifically, three Na atoms in the Na 4 tetrahedron are each situated on a B-B edge. The GM sandwich cluster may be referred to as the staggered conformation.
Twisting the Na 4 tetrahedron against the B 7 molecular wheel by 30 • , one readily reaches another C 3v ( 1 A 1 ) structure, which turns out to be a transition state (TS), as depicted in Figure 1b. The TS structure is referred to herein as the eclipsed conformation. The closest low-lying structure, LM C 2v ( 1 A 1 ), is also a sandwich ( Figure 2). It differs from the GM structure in the top Na 4 ligand. The Na 4 ligand in the LM cluster is distorted into a roof-like (or rhombic) structure, so that only two Na atoms are coordinated to the B-B edges. The distortion alters the nature of bonding in the Na 4 unit (vide infra).
Among the low-lying isomers is a local-minimum (LM) C2v ( 1 A1) structure S1), which is about 0.2 eV above the GM cluster. The remaining isomers are subst higher in energy. Considering that adequate intramolecular charge transfers ca place from Na to B, the triplet-state structures are relatively unimportant for the cluster system with an even number of 26 valence electrons. Indeed, our calculation that the corresponding triplet-state structures for the top 12 singlet-state isomers in S1 are 0.82-2.20 eV higher than the GM cluster at the PBE0/6-311 + G* level. Spec as illustrated in Figure S2, the lowest-lying triplet-state geometry of Na5B7 cluster, lies 0.82 eV higher above the GM structure. Other triplet-state geometries are even in energy, by up to 2.20 eV (relative to the singlet GM cluster), for the 12th isomeric structure. Thus, the C3v ( 1 A1) cluster ( Figure 1a) is reasonably well defined on its po energy surface as the real GM structure. Alternative structures, either singlets or t are not energetically competitive, except for an LM C2v ( 1 A1) structure as mentioned The top-and side-views of GM C3v ( 1 A1) Na5B7 cluster are presented in Fig  Basically, it is a three-layered sandwich cluster. The boron component forms a qu nar B7 molecular wheel as the core layer of the sandwich. It features six-fold in-pl ordination for the central B site. On the other hand, the Na5 component is divided Na4 tetrahedron and an isolated Na atom. The two Na-based layers serve as liga the B7 molecular wheel. Specifically, three Na atoms in the Na4 tetrahedron are ea ated on a B-B edge. The GM sandwich cluster may be referred to as the staggered mation.
Twisting the Na4 tetrahedron against the B7 molecular wheel by 30°, one reaches another C3v ( 1 A1) structure, which turns out to be a transition state (TS), as d in Figure 1b. The TS structure is referred to herein as the eclipsed conformation. The low-lying structure, LM C2v ( 1 A1), is also a sandwich ( Figure 2). It differs from t structure in the top Na4 ligand. The Na4 ligand in the LM cluster is distorted into like (or rhombic) structure, so that only two Na atoms are coordinated to the B-B The distortion alters the nature of bonding in the Na4 unit (vide infra).

Bond Distances, Wiberg Bond Indices, and Natural Atomic Charges
The calculated bond distances, Wiberg bond indices (WBIs), and natural charges for GM and TS structures of the Na5B7 cluster are shown in Figures S3, 3, respectively. The three sets of data are generally coherent with each other. In par the GM and TS structures are quite similar, either qualitatively or quantitatively. fore, we shall primarily describe the GM cluster only.

Bond Distances, Wiberg Bond Indices, and Natural Atomic Charges
The calculated bond distances, Wiberg bond indices (WBIs), and natural atomic charges for GM and TS structures of the Na 5 B 7 cluster are shown in Figures S3, 3 and S4, respectively. The three sets of data are generally coherent with each other. In particular, the GM and TS structures are quite similar, either qualitatively or quantitatively. Therefore, we shall primarily describe the GM cluster only.
In the GM cluster, the B 7 wheel has uniform B-B distances for the peripheral links (1.61 Å) and radial ones (1.64 Å). These distances are shorter than the single bond (upper limit: 1.70 Å) [33]. It is invaluable to compare these with bare B 7 cluster, whose peripheral and radial distances are 1.56-1.62 and 1.69-1.76 Å, respectively [34]. Thus, the peripheral B 6 ring in GM Na 5 B 7 cluster seems to be expanded, despite the fact that the radial B-B distances are clearly shortened. The above structural data are in line with a negatively charged B 7 wheel in GM Na 5 B 7 cluster.
B6 ring in GM Na5B7 cluster seems to be expanded, despite the fact that the radial B-B distances are clearly shortened. The above structural data are in line with a negatively charged B7 wheel in GM Na5B7 cluster.
The Na-Na distances for Na4 tetrahedron in the GM structure are 3.60/3.64 Å, whose shape also differs distinctly from that of a bare Na4 cluster. The latter assumes a rhombic D2h structure [35]. The above Na-Na distances are far longer than a Na-Na single bond (3.10 Å) [33]. The tetrahedral Na4 ligand is not in a neutral state (vide infra). The shortest B-Na distance is 2.51 Å, which is longer than the recommended value for a B-Na σ single bond. The calculated WBIs fully support the above assignments (Figure 3a). The peripheral and radial B-B links have WBI values of 1.35 and 0.63, respectively. The Na-Na edges in the Na4 ligand have WBIs of 0.14-0.32, which are collectively consistent with a delocalized four-center two-electron (4c-2e) σ bond. According to the above analysis, sandwich Na5B7 cluster has quite substantial intramolecular charge-transfers in between the B7 core and the Na4/Na ligands. The calculated natural atomic charges offer a quantitative picture ( Figure S4a). Basically, each B atom in the B7 wheel has a negative charge from −0.27 to −0.35 |e|. Overall, the B7 wheel carries a total charge of −2.37 |e|. The tetrahedral Na4 ligand has a collective charge of +1.49 |e|, whereas the Na ligand carries a charge of +0.87 |e|. Clearly, the sandwich cluster can be formulated as a charge-transfer [Na4] 2+ [B7] 3− [Na] + complex. A similar analysis of the structure, WBIs, and natural atomic charges of the LM structure of Na5B7 cluster may be performed (see Figure S5) . We have also run structural optimization for a bare B7 3− trianion cluster at the PBE0/6-311 + G* level. The trianionic nature observed suggests that this should at most be considered as a model cluster. The interatomic distances may tend to expand to some extent, due to intramolecular Coulomb repulsion associated with three extra charges. The calculated The Na-Na distances for Na 4 tetrahedron in the GM structure are 3.60/3.64 Å, whose shape also differs distinctly from that of a bare Na 4 cluster. The latter assumes a rhombic D 2h structure [35]. The above Na-Na distances are far longer than a Na-Na single bond (3.10 Å) [33]. The tetrahedral Na 4 ligand is not in a neutral state (vide infra). The shortest B-Na distance is 2.51 Å, which is longer than the recommended value for a B-Na σ single bond. The calculated WBIs fully support the above assignments ( Figure 3a). The peripheral and radial B-B links have WBI values of 1.35 and 0.63, respectively. The Na-Na edges in the Na 4 ligand have WBIs of 0.14-0.32, which are collectively consistent with a delocalized four-center two-electron (4c-2e) σ bond.
According to the above analysis, sandwich Na 5 B 7 cluster has quite substantial intramolecular charge-transfers in between the B 7 core and the Na 4 /Na ligands. The calculated natural atomic charges offer a quantitative picture ( Figure S4a). Basically, each B atom in the B 7 wheel has a negative charge from −0.27 to −0.35 |e|. Overall, the B 7 wheel carries a total charge of −2.37 |e|. The tetrahedral Na 4 ligand has a collective charge of +1.49 |e|, whereas the Na ligand carries a charge of +0.87 |e|. Clearly, the sandwich cluster can be formulated as a charge-transfer [Na 4 ] 2+ [B 7 ] 3− [Na] + complex. A similar analysis of the structure, WBIs, and natural atomic charges of the LM structure of Na 5 B 7 cluster may be performed (see Figure S5).
We have also run structural optimization for a bare B 7 3− trianion cluster at the PBE0/6-311 + G* level. The trianionic nature observed suggests that this should at most be considered as a model cluster. The interatomic distances may tend to expand to some extent, due to intramolecular Coulomb repulsion associated with three extra charges. The calculated bond distances and WBIs of the model B 7 3− disk cluster are shown in Figure S6. The bond distances of the peripheral B-B links and radial ones are only 0.02 and 0.04 Å longer than those in the GM Na 5 B 7 cluster, respectively. The WBIs of the peripheral B-B links and the radial ones show very minor variations relative to those in GM Na 5 B 7 cluster, by 0.03 and 0.02, respectively. The central and peripheral B atoms in the model B 7 3− cluster have negative charges of −0.30 and −0.45 |e|, respectively, which are close to those in the GM Na 5 B 7 cluster. Overall, the central B atom in model B 7 3− cluster is slightly above the B 6 ring plane by 0.40 Å, as compared to 0.26 Å in GM Na 5 B 7 cluster. The sandwich GM Na 5 B 7 cluster moderately flattens the B 7 wheel, which is intuitively expected and not surprising. The above comparison between the bare B 7 3− model cluster and the B wheel in Na 5 B 7 cluster further confirms the [B 7 ] 3− nature of the B wheel in GM Na 5 B 7 cluster. It is stressed that the [B 7 ] 3− nature in GM Na 5 B 7 cluster represents a solid conclusion on the basis of the structural data, natural bond orbital (NBO) analysis, and in particular the canonical molecular orbital (CMO) analysis and adaptive natural density partitioning (AdNDP) results. This conclusion stands firmly even without any calculations on a bare B 7 3− model cluster. The latter calculations are merely an extra computational effort, which is a secondary part in the present paper.

Methods
The GM structure and low-lying isomers of Na 5 B 7 cluster were explored by computer global searches using the Coalescence Kick (CK) algorithm [36,37], which were also aided by manual structure constructions. About 3000 stationary points were probed on the potential energy surface. The Gaussian 09 program was used subsequently to reoptimize the structures at the PBE0/6-311 + G* level [38][39][40]. To verify the reliability in terms of energetics, the top candidate structures were further assessed at the B3LYP/6-311 + G* and single-point CCSD(T)/6-311 + G*//PBE0/6-311 + G* levels [41]. Vibrational frequencies were calculated at the same density-functional theory (DFT) levels, that is, PBE0 and B3LYP, to ensure that the reported low-lying structures are true minima on the potential energy surface of the system, unless specifically stated otherwise.
Chemical bonding was elucidated using the CMO analysis, as well as the AdNDP analysis [42]. The AdNDP results were visualized using the Molekel 5. 4.0.8 program [43]. The WBIs and charge distribution were calculated by NBO analysis [44] at the PBE0/6-311 + G* level. Born-Oppenheimer molecular dynamics (BOMD) simulations [45] were performed at PBE0/6-31G to demonstrate the structural fluxionality of the system.

Chemical Bonding
An in-depth chemical bonding analysis is essential toward understanding the stability, unique structure, and dynamic fluxionality of the title Na 5 B 7 cluster. The CMO analysis is fundamental for this purpose. The Na 5 B 7 cluster has 26 valence electrons. Their occupied CMOs are depicted in Figure 4. Of these 13 CMOs, six σ CMOs in subset (a) are primarily composed of B 2s atomic orbitals (AOs) of the peripheral B 6 ring. These CMOs show from 0, 1, 2 to 3 nodal planes, including two degenerate pairs in the middle. Following the orbital construction principles, they can be recombined and localized as six Lewis-type two-center, two-electron (2c-2e) σ single bonds, one for each B-B edge.
Subset (b) in Figure 4 shows three π CMOs on the B 7 wheel. Owing to the six-fold symmetry of the wheel, this π sextet cannot be localized as Lewis-type π bonds, akin to that in benzene, thus rendering π aromaticity for the sandwich cluster. The magic 6π electron counting conforms to the (4n + 2) Hückel rule. Likewise, the three σ CMOs in subset (c) have similar spatial distribution compared to those in the π sextet, except that the former CMOs are σ in nature. Again, the σ sextet is intrinsically delocalized and cannot be reduced to Lewis-type σ single bonds. It is, therefore, imperative to claim σ aromaticity for the sandwich as well, following the (4n + 2) Hückel rule.
Lastly, subset (d) shows only one σ CMO, which is clouded on the Na 4 ligand. It is 4c-2e in nature and cannot be localized as one Lewis-type Na-Na σ single bond. The delocalized nature of these 2σ electrons renders three-dimensional σ aromaticity for the tetrahedral Na 4 ligand and, hence, the sandwich cluster. Overall, the sandwich Na 5 B 7 cluster collectively features three-fold π/σ aromaticity with the 6π/6σ/2σ electron counting.
The bonding picture is perfectly borne out from the AdNDP analysis. The AdNDP scheme of sandwich GM Na 5 B 7 cluster is presented in Figure 5. The above analysis suggests that there is relatively minor covalent bonding between the three layers of the sandwich, and the cluster is indeed a charge-transfer complex. Similar CMO and AdNDP analyses can be conducted for the TS Na 5 B 7 cluster, which are relatively straightforward and easy to understand. The relevant data are presented in Figures S7 and S8.  Figure 4 shows three π CMOs on the B7 wheel. Owing to the six-fold symmetry of the wheel, this π sextet cannot be localized as Lewis-type π bonds, akin to that in benzene, thus rendering π aromaticity for the sandwich cluster. The magic 6π electron counting conforms to the (4n + 2) Hückel rule. Likewise, the three σ CMOs in subset (c) have similar spatial distribution compared to those in the π sextet, except that the former CMOs are σ in nature. Again, the σ sextet is intrinsically delocalized and cannot be reduced to Lewis-type σ single bonds. It is, therefore, imperative to claim σ aromaticity for the sandwich as well, following the (4n + 2) Hückel rule.

Subset (b) in
Lastly, subset (d) shows only one σ CMO, which is clouded on the Na4 ligand. It is 4c-2e in nature and cannot be localized as one Lewis-type Na-Na σ single bond. The delocalized nature of these 2σ electrons renders three-dimensional σ aromaticity for the tetrahedral Na4 ligand and, hence, the sandwich cluster. Overall, the sandwich Na5B7 cluster collectively features three-fold π/σ aromaticity with the 6π/6σ/2σ electron counting.
The bonding picture is perfectly borne out from the AdNDP analysis. The AdNDP scheme of sandwich GM Na5B7 cluster is presented in Figure 5. The above analysis suggests that there is relatively minor covalent bonding between the three layers of the sandwich, and the cluster is indeed a charge-transfer complex. Similar CMO and AdNDP analyses can be conducted for the TS Na5B7 cluster, which are relatively straightforward and easy to understand. The relevant data are presented in Figures S7 and S8.
As shown in Figure 6, the GM and TS geometries of sandwich Na 5 B 7 cluster are connected straightforwardly. Starting from GM (labeled as "GM 1 ") and twisting the B 7 wheel clockwise relative to the Na 4 tetrahedron by 30 • , one readily reaches the TS structure. Further rotation in the same direction by 30 • results in recovery of the GM structure (labeled as "GM 2 "). The energy barrier between the GM and TS structures is 0.04 eV at the PBE0/6-311 + G* level, which is relatively minor considering the anticipated strong electrostatic interaction between the B 7 wheel and the Na 4 /Na ligands in the sandwich. The calculated soft vibrational modes of 45.8   To further validate the dynamic fluxionality of Na5B7 cluster, a BOMD simulation was performed at a selected temperature of 300 K. Specifically, the BOMD simulation was performed using the Hessian-based predictor-corrector method [48] at an initial setup temperature of 300 K, which was run for a time span of 50 ps (10,000 steps) at the PBE0/6-31G level. We took the GM geometry as the initial structure and simulated the dynamic evolution process. The vibrational sampling temperature was shown to be 298 K, which  To further validate the dynamic fluxionality of Na5B7 cluster, a BOMD simulation was performed at a selected temperature of 300 K. Specifically, the BOMD simulation was performed using the Hessian-based predictor-corrector method [48] at an initial setup temperature of 300 K, which was run for a time span of 50 ps (10,000 steps) at the PBE0/6-  To further validate the dynamic fluxionality of Na 5 B 7 cluster, a BOMD simulation was performed at a selected temperature of 300 K. Specifically, the BOMD simulation was performed using the Hessian-based predictor-corrector method [48] at an initial setup temperature of 300 K, which was run for a time span of 50 ps (10,000 steps) at the PBE0/6-31G level. We took the GM geometry as the initial structure and simulated the dynamic evolution process. The vibrational sampling temperature was shown to be 298 K, which is close to the setting temperature. Before being recalculated analytically, the Bofill update method was used to update the Hessian evaluation for five steps. A trajectory step size of 1.0 amu 1/2 bohr was used in the whole simulation process, and the total number of trajectories was 1. The maximum point for each trajectory was 200,000. Despite the variation in potential and kinetic energies during the BOMD process, the total energy remained constant, which was conserved to 10 −6 hartree during the simulation process. A short movie extracted from the BOMD data is presented in the Supplementary Materials, which vividly demonstrates the intriguing structural dynamics of the cluster (see Video S1). In short, the title cluster is dynamically fluxional even at near room temperature. We comment here that the BOMD simulation represents a relatively minor part in this study. It is not intended to provide any information with accuracy. It offers molecular dynamics information of the system qualitatively, rather than quantitatively. 4. 3. On the Low-Lying LM Structure: The Importance of Three-Fold π/σ Aromaticity for GM Na 5 B 7 Cluster Our exploration of the potential energy surface of Na 5 B 7 cluster shows that only two structures, GM (C 3v , 1 A 1 ) and LM (C 2v , 1 A 1 ), are close in energy, within about 0.2 eV at all three levels of theory ( Figure S1). Other isomeric structures are substantially higher in energy, by at least 0.9 eV. Therefore, it is invaluable to understand the LM cluster, which is depicted in Figure 2. Basically, the LM and GM structures differ in the upper Na 4 ligand. While the Na 4 ligand in GM cluster is a slightly distorted tetrahedron (3.60 versus 3.64 Å; Figure S3a), it becomes a roof-like, quasi-planar ligand (or a rhombic ligand) in the LM cluster ( Figure S5a). In the LM cluster, the roof-like Na 4 ligand appears to interact with the B 7 wheel primarily via two Na atoms along the longer diagonal, which allows optimal intramolecular charge transfer ( Figure S5c). As a consequence, chemical bonding in the roof-like Na 4 ligand is dominated by σ bonding along the shorter diagonal. The shorter Na 2 diagonal in the highest occupied molecular orbital (HOMO) contributes to 66% of the whole Na 4 ligand (Figures S9d and S10b). In other words, chemical bonding in the roof-like Na 4 ligand is largely a Lewis-type 2c-2e σ bond in nature, at least in the zeroth-order picture.
In contrast, the 2σ framework in GM cluster is truly delocalized in a three-dimensional manner, which offers extra stabilization to the sandwich GM cluster. As a model system, one can evaluate a free-standing Na 4 2+ dication cluster in its tetrahedral T d and rhombic D 2h geometries ( Figure S11). At the PBE0 level, the T d cluster is 0.21 eV more stable than its rhombic D 2h isomer. This energetic difference is comparable to those between the GM and LM structures of Na 5 B 7 cluster ( Figure S1). Thus, 2σ aromaticity is clearly a decisive factor that helps distinguish between the GM Na 5 B 7 cluster from its LM isomer.
The tetrahedral Na 4 ligand, in its specific [Na 4 ] 2+ charge state, is the simplest polyhedral structure. Hence it can be considered the simplest example of "spherical" aromaticity. The 2σ electron counting also conforms to the 2(n + 1) 2 rule for spherical aromaticity. Alternatively, the tetrahedral Na 4 ligand may be viewed as the simplest superatom [49][50][51].
Interestingly, one can directly connect the GM and LM structures via a TS geometry, TS LM-GM , as shown in Figure 8. The TS LM-GM is located at only 0.13 eV above LM at PBE0. The TS LM-GM structure has a soft imaginary mode of 53.0i cm −1 . The GM cluster lies 0.19 eV below LM, as well as 0.32 eV below TS LM-GM . According to this energy diagram, the GM cluster is relatively robust against isomerization. Our electronic structure calculations also indicate that the energy gap of GM Na 5 B 7 cluster (2.89 eV; see Figure 9), between its HOMO and lowest unoccupied molecular orbital (LUMO), is significantly larger than that in its LM isomer (0.25 eV), thus demonstrating the electronic robustness of sandwich GM Na 5 B 7 cluster. Indeed, the sizable energy gap in Figure 9 is a strong indication that a [B 7 ] 3− motif in binary Na 5 B 7 cluster is appropriate, which favors a closedshell configuration in the triply charged anionic state. The same reason underlies the nature of a charge-transfer [Na 4 ] 2+ [B 7 ] 3− [Na] + complex for GM Na 5 B 7 cluster, as well as its unique sandwich geometry.

Conclusions
We have computationally designed a boron-based alloy Na5B7 cluster that assumes unique three-layered sandwich structure. The cluster may be described as a charge-transfer [Na4] 2+ [B7] 3− [Na] + complex, which is composed of a quasi-planar B7 wheel as core, as well as a Na4 tetrahedron and an isolated Na atom as its two ligand layers at the top and

Conclusions
We have computationally designed a boron-based alloy Na5B7 cluster that assumes unique three-layered sandwich structure. The cluster may be described as a charge-transfer [Na4] 2+ [B7] 3− [Na] + complex, which is composed of a quasi-planar B7 wheel as core, as well as a Na4 tetrahedron and an isolated Na atom as its two ligand layers at the top and

Conclusions
We have computationally designed a boron-based alloy Na 5 B 7 cluster that assumes unique three-layered sandwich structure. The cluster may be described as a charge-transfer [Na 4 ] 2+ [B 7 ] 3− [Na] + complex, which is composed of a quasi-planar B 7 wheel as core, as well as a Na 4 tetrahedron and an isolated Na atom as its two ligand layers at the top and at the bottom. Chemical bonding analysis indicates magic 6π/6σ double aromaticity for the [B 7 ] 3− wheel and three-dimensional 2σ aromaticity for the tetrahedral [Na 4 ] 2+ ligand. The latter is also the simplest example of three-dimensional or spherical aromaticity, as well as the simplest superatom. Collectively, the Na 5 B 7 cluster has three-fold 6π/6σ/2σ aromaticity, which underlies its thermodynamic stability. The same mechanism facilitates the intriguing dynamic structural fluxionality for the sandwich cluster, even at near room temperature. This work also highlights the structural and electronic robustness of the [B 7 ] 3− molecular wheel as a potential inorganic ligand.
Supplementary Materials: The following supporting information can be downloaded at: https://www.mdpi.com/article/10.3390/molecules28073276/s1. Table S1: Cartesian coordinates for global-minimum (GM) and transition-state (TS) structures of Na 5 B 7 cluster at PBE0/6-311 + G*; Figure S1: Alternative optimized structures of Na 5 B 7 cluster at PBE0; Figure S2: Selected triplet-state structures of Na 5 B 7 cluster; Figures S3 and S4: Calculated bond distances and natural atomic charges for GM and TS structures of Na 5 B 7 cluster; Figure S5: Bond distances, Wiberg bond indices (WBIs), and natural atomic charges for a local minimum (LM) structure; Figure S6: Calculated bond distances and WBIs of model B 7 3− cluster at PBE0/6-311 + G*; Figures S7-S10: CMOs and AdNDP schemes of TS and LM structures of Na 5 B 7 cluster; Figure S11: Relative energies for tetrahedral versus rhombic structures of a free-standing Na 4 2+ model cluster at PBE0; Video S1: A short movie extracted from the BOMD simulation for Na 5 B 7 cluster at 300 K. Institutional Review Board Statement: Not applicable.