1 Introduction

Boron and its compounds have attracted considerable interests owing to their unusual chemical bonding and novel chemical/physical properties [1,2,3,4,5,6]. With unique nature of electron deficiency, B atom can bond with most elements via a favorable delocalized multicenter bond. To date, many experimental and computational investigations revealed the diverse structural patterns of pure Bn clusters, such as planar, quasi-planar, bilayer, tubular, cage-like and core–shell structures [6,7,8,9,10].

The small-sized Bn clusters (n < 20) are dominated by planar or quasi-planar configurations, which are stabilized by electron delocalization in both σ and π frameworks. Among them, highly symmetrical wheel-like B7−, B8− and B9− are distinctively characterized by the presence of a central B atom [11, 12], which further motivate systematic explorations on substitutional doping of the central B atom [13,14,15,16,17,18]. Ito et al. predicted several planar hypercoordinate CoB8−, FeB9− and FeB82− clusters, which are stabilized by “double aromatic” character with six π electrons and ten radical electrons [19, 20]. These cluster species were later confirmed by the experimental photoelectron spectroscopic studies by Wang and coworkers [16, 17]. Wang’s group also proposed an electronic design principle based on the double aromaticity requirement to achieve stable transition metal-centered planar Bn clusters [16]. Following this principle, many clusters have been obtained, such as RuB90/−, RhB90/−, IrB90/−, Re9−, TaB10−, NbB10−, etc. [21,22,23]. So far, the planar metal-centered Bn clusters with highest coordination number are TaB10− and NbB10− [22]. Considering the proper atomic sizes and number of valence electrons, the transition metal (M) elements with partially filled d orbitals are more suitable as the central dopant to stabilize a Bn wheel by forming delocalized bonds with B atoms.

In addition to the metal-centered wheel-like structures, several other structural motifs were also identified in small-size metal-doped Bn clusters. A typical one is the half-sandwich structure, which is a bipyramid with a metal atom located at the apical site. Liu et al. [24] investigated 3d transition metal-doped Bn clusters and found n = 7 marks the onset of the half-sandwich structure, among which CrB7, FeB7 and CoB7 clusters possess relatively higher stabilities. Later, AlB7, AlB8 and PrB7 were featured as an metal atom interacting ionically with a hexagonal B7 (C6v) or heptagonal B8 (C7v) pyramidal fragment [14, 25]. As the B73− and B82− molecular wheels satisfy double aromaticity, ligands bonding with a metal dopant further enhance their stability. When n < 7, the metal dopant prefers staying at the edges of Bn clusters [24, 26, 27]. For instance, MB6 favors a hexagonal geometry with a B atom in the center and the metal atom located at the edge site [28, 29]. For n ≥ 10, most metal-doped Bn clusters adopt bowl-like or drum-shaped structures [30,31,32,33].

Despite many reports on the metal-doped boron clusters, little is known about the small-size Bn clusters doped with a late transition metal atom. Besides the distinctive wheel-like and half-sandwich structures, are there any other structural patterns in this size range? What is the bonding character between the d orbitals of the transition metal atom and the 2 s and 2p orbitals of B atoms? In this work, we present a systematic study on 4d and 5d transition metal-doped boron clusters in order to elucidate the structural patterns and bonding characters of MBn (n = 7–10). In addition to the known wheel-like and half-sandwich structural patterns, an umbrella-like structure is found in AgB9 and AuB9 for the first time. Among all the considered clusters, the half-sandwich RhB7, IrB7, RuB8 and OsB8 and umbrella-like AgB9 and AuB9 clusters exhibit higher stability, which can be correlated to their closed electron shells following a Jellium model of the circular disk.

2 Methods

Unbiased global search of MBn (n = 7–10) clusters was carried out using our own comprehensive genetic algorithm (CGA) code [34] incorporated with density functional theory (DFT) calculations [35] (CGA-DFT). For each cluster, CGA search was conducted with at least 1000 iterations for a population of 12 members to ensure the global minimum structure within the used computational approach. The child clusters from crossover and optional mutation operations were fully relaxed using the double numerical basis set plus d-polarization functions (DND) combined with the Perdew–Burke–Enzerhof (PBE) functional within the generalized gradient approximation (GGA) [36, 37], as implemented in the DMol3 program [35]. The validity and efficiency of the present CGA-DFT scheme have been well demonstrated in our recent studies on B [10, 38] and doped B [39, 40] clusters. Further details of CGA can be found in the review articles [34, 41].

The low-lying isomers from CGA-DFT search were further optimized by considering different spin multiplicities at higher accuracy, with the PBE0 functional [42], and 6-311 + G(d) and LANL2DZ basis sets for B and M atoms, respectively. To confirm the PBE0 results, the relative energies between some selected isomers were further calculated at the CCSD(T) level with the same basis sets [43]. By computing the vibrational frequencies of these MBn clusters, we verified that there is no imaginary frequency corresponding to the saddle point on the potential energy surface, and we added the zero-point-energy (ZPE) to the total energy of each isomer. The energetic stability of a MBn cluster is characterized by the doping energy (Ed) of M atom defined as:

$$ E_{{\text{d}}} \left( {MB_{n} } \right) = E\left( {B_{n} } \right) + E\left( M \right) - E\left( {MB_{n} } \right), $$
(1)

where E(MBn) is energy of the MBn cluster, E(Bn) is the energy of the Bn moiety in MBn without structural relaxation, and E(M) is the energy of an individual M atom, respectively. Chemical bonding analyses were performed using the natural bond orbital (NBO) and adaptive natural density partitioning (AdNDP) methods [44, 45]. The nucleus-independent chemical shift (NICS) [46] and zz NICS shielding tensor components (NICSzz) [47] were computed to assess the aromaticity. Except for CGA-DFT search, all of the above calculations and analyses were performed using the Gaussian 09 package [48].

3 Results and discussion

3.1 Global minimum structures of MBn clusters

The ground-state structures of MBn clusters (n = 7−10) from our high-throughput structural searches are presented in Fig. 1 and Fig. S1–S4 of Supporting Information. All these 72 cluster structures can be classified into half-sandwich, wheel-like, umbrella-like, and quasi-planar structural patterns. Among them, the umbrella-like one has never been reported before.

Fig. 1
figure 1

Structural patterns of MBn (n = 7‒10) clusters. The tilde symbol ( ~) between Y and Rh, W and Ir, Nb and Rh means that the elements belonging to the same period of the periodic table between those two elements (e.g., Y ~ Rh means Y, Zr, Nb, Mo, Tc, Ru, Rh)

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For 4d transition metals, the MBn clusters doped with early and middle M elements adopt half-sandwich structures with the quasi-planar B7 (C6v), B8 (C7v), B9 (Cs) and B10 (C2v) moieties coordinated to the M atom, while the late M-doped Bn clusters usually prefer quasi-planar Bn configuration with the M atom located at the low-coordination peripheral position. As exceptional cases, the lowest energy structures of RhB9 and PdB9 are formed by an RhB2 trimer and a PdB dimer bonded with quasi-planar B7 and B8 moieties, respectively. Interestingly, the ground state of AgB9 is an unprecedented umbrella-like structure with C7v symmetry. It is constructed by an Ag–B heteronuclear dimer bonded with the B8 unit (C7v), with the closed d-shell Ag atom having low coordination.

Similar to the cases of 4d transition metal doping, the 5d transition metal-doped Bn clusters adopt hypercoordinated half-sandwich or wheel-like structures for early and middle M atoms and lower-coordinated quasi-planar or umbrella-like structures for late M atoms, respectively. Roughly speaking, the half-sandwich pattern dominates the ground states for most considered MBn clusters. The M dopant atoms prefer the high-coordinated center of half-sandwich, except TaB7 having the Ta atom replacing a B atom on the peripheral ring of B7 framework. As the cluster size increases, the first wheel-like structure occurs at the critical size n = 8, among which the quartet spin state of ReB8 agrees with the previous report by Chen et al. [23] and the doublet spin state of IrB8 is found for the first time. Noticeably, a half-sandwich OsB8 cluster with singlet electronic state emerges between the wheel-like structural pattern (ReB.8 and IrB8). Thus, not only the cluster size but also the specific electronic configuration play more important roles in determining the ground-state structure of M-centered Bn clusters.

Among the considered MBn clusters, the highest coordination number in the wheel-like species is nona-coordinated OsB9 and IrB9 clusters with C9v symmetry. IrB9 has a singlet spin multiplicity and has been reported by Li et al. [21], while OsB9 with doublet spin state is reported for the first time. However, the wheel-like ReB9 structure suggested by Chen et al. [23] is higher in energy by 0.39 eV than the half-sandwich one according to our PBE0 calculation. Generally speaking, 5d M atoms are suitable central dopants in the eight- and nine-membered rings of boron, such as the cases of ReB8, IrB8, OsB9 and IrB9, while the isoelectronic 4d M atoms are too small to stabilize the B8 and B9 wheels. The umbrella-like pattern only occurs for MB9 clusters doped with the group IB elements, i.e., AgB9 and AuB9.

With increasing number of B atoms, the growth behavior of MBn clusters can be viewed as gradually adding B atom on the quasi-planar skeleton of Bn. Some exceptional cases are wheel-like and umbrella-like patterns in the middle or late transition metal-doped Bn clusters, respectively. Up to n = 10, the preferred structural pattern becomes relatively simpler, i.e., either C2v half-sandwich or C1 quasi-planar structure. The B10 ligand framework is a fragment of triangular lattice similar to the bare B10 cluster, in which two interior B atoms are surrounded by eight peripheral B atoms [49].

3.2 Stabilities of MBn clusters

To discuss the relative stability of MBn (n = 7−10) clusters, the average binding energy (Eb) and HOMO–LUMO gap (EHL) are plotted in Fig. 2 for all the 72 clusters. Intuitively, those clusters occupying the upper right region (highlighted by the dotted circle in Fig. 2) possess relatively higher thermodynamic and chemical stabilities. Among them, the half-sandwich RhB7, IrB7, RuB8 and OsB8; wheel-like IrB9; and umbrella-like AgB9 and AuB9 clusters stand out with large HOMO–LUMO gap (EHL > 3.4 eV) and appreciable binding energy (Eb > 4.5 eV) owing to their closed electronic shell. Our CCSD(T) calculations further confirm that these MBn cluster structures are indeed the ground states (see Table S1). To further support the results about the first reported umbrella-like AgB9 and AuB9 clusters, we examined the adiabatic electron affinity (AEA) and the relative energies of low-lying isomers of them using a variety of common functionals including HSE06, B3LYP, B3PW91 and TPSSh. The umbrella-like AgB9‒I and AuB9‒I with the smallest AEA are always the ground-state structures calculated with different functionals (see Table S2). In the following contents, we will focus on these seven relatively stable clusters.

Fig. 2
figure 2

Scatter plots of relative stability of MBn (n = 7‒10) clusters. Shapes and colors of the data points indicate the cluster sizes n and type of M elements, respectively. The clusters with higher binding energy (Eb), larger HOMO–LUMO gap (EHL) and higher symmetry are enclosed by the dotted circles, and the structures of clusters with closed electronic shell are shown as insets

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Furthermore, the energetic stability of the above clusters can be characterized by the doping energy (Ed) of M atom defined in Eq. (1). As shown in Table 1, the clusters with higher coordination have larger doping energy. For instance, IrB9 cluster with nona-coordination on Ir has a giant Ed of 13.18 eV, while AgB9 cluster with lower coordination of M atom has an Ed of only 2.41 eV. Moreover, the doping energy of the 4d-M-doped cluster is slightly larger than the isoelectronic 5d-M-doped cluster within the same half-sandwich pattern. In addition, the total Wiberg bond indexes (WBI) of the M atom (WBIM) range from 0.86 (AgB9) to 4.42 (OsB8), and the total WBI of 5d-M is larger than that of 4d-M in the same structural pattern. All these results imply that the 5d-M atom has stronger interaction with the Bn moiety than that of 4d-M. Meanwhile, the average WBI of M−B(peripheral) (i.e., WBIM−B1) is larger than that of M−B(central) (i.e., WBIM−B2), suggesting that the M atom bonds stronger with peripheral B atoms than with central B atom in the half-sandwich pattern. Among them, the OsB8 cluster has the largest WBIM and WBIM−B1 values, associated with the largest binding energy of 5.15 eV. In the umbrella-like AgB9 and AuB9 clusters, the WBI M−B2 values are 0.67 and 0.96, respectively, reflecting appreciable interactions between the M atom and the adjacent B atoms.

Table 1 Binding energy per atom (Eb), HOMO–LUMO gap (EHL), doping energy (Ed), and Wiberg bond indexes (WBIM, WBIM−B1 and WBIM−B2) of M total bond order, average M–B(peripheral) bond and M–B(central) bond of MBn clusters
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For a cluster or molecule, the vertical ionization potential (VIP) and vertical electron affinity (VEA) are characteristic electronic properties that describe the capability to loss or gain an electron, while the difference between VIP and VEA known as the chemical hardness [50, 51] is a useful metric for estimating the stability of the cluster. In addition, the adiabatic electron affinity (AEA) is also calculated to evaluate the chemical stability of cluster, which is defined as the binding energy of an electron to the neutral cluster. All these values for the seven highly stable MBn (n = 7−10) clusters are summarized in Table 2. The chemical hardness for these clusters are relatively large, i.e., ranging from 6.51 eV (IrB7) to 8.07 eV (AuB9), much larger than those of other MBn clusters with singlet state (i.e., 5.43 eV and 6.03 eV for half-sandwich YB7 and LaB7, 6.52 eV and 4.92 eV for quasi-planar AgB7 and PtB8, respectively, shown in Table S3). Meanwhile, these values are larger than that of the typical 2D closed-shell cluster like Au6 (6.52 eV) and comparable to those of the larger-size metal-capsulated boron cages W@B24 and Mo@B24 with 18-electron closed shell (7.13 and 6.81 eV). Moreover, the differences of VIP and VEA of 5d-M dopants are larger than those of 4d-M dopants within the same structural pattern implying that the 5d-M-doped Bn clusters with larger chemical hardness would possess higher stability. The relatively small AEA values mean high stability of those closed-shell clusters and are consistent with their large HOMO–LUMO gap. In addition, NBO analyses unveil that the M atom in the wheel-like pattern possesses negative charge and thus acts as an electron acceptor, while it holds positive charge in umbrella-like pattern and generally remains electric neutrality in the half-sandwich pattern.

Table 2 Vertical ionization potentials (VIP), vertical electron affinities (VEA), adiabatic electron affinities (AEA), difference between VIP and VEA, and natural population analysis (NPA) charges of M atoms, and NICSzz values corresponding to NICS(0)zz at the geometry center (NICS(1)zz at a point 1.0 Å above the ring center for IrB9 only)
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Compared to the umbrella-like AgB9 (AuB9) cluster, the quasi-planar Cs AgB8 and C1 AgB10 (AuB8 and AuB10) with an Ag (Au) atom bound at the edge show relatively low stability. Intuitively, this is because the quasi-planar ones have lower symmetry. Meanwhile, with open electronic configuration, the quasi-planar ones have smaller HOMO–LUMO gap and larger AEA, and the chemical hardness for these clusters are relatively weak (see Table S4), i.e., ranging from 4.79 eV (AuB10) to 5.46 eV (AgB8), much smaller than 7.38 eV (AgB9) and 8.07 eV (AuB9). Furthermore, the AdNDP analyses reveal a single electron occupying one π bond in AgB8 (AuB8) cluster, which deviates from the 4 N + 2 Hückel rule of aromaticity (see Fig. S5 a–d). The detailed information about the bonding nature will be discussed in Sect. 3.3.

3.3 Chemical bond analysis

Based on the analyses of relative stability, we further choose half-sandwich IrB7 and RuB8, wheel-like IrB9 and umbrella-like AgB9 clusters as examples to probe the bonding nature between the M atom and the Bn moiety by AdNDP analyses (Fig. 3). The clusters with identical structural pattern are shown in Fig. S6.

Fig. 3
figure 3

AdNDP chemical bonding patterns of a IrB7, b RuB8, c IrB9, and d AgB9 clusters with the occupation number (ON)

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For the half-sandwich IrB7 and RuB8 clusters, the AdNDP analyses [52, 53] reveal 2c–2e σ bonds on the B−B bond of circumference, including six σ bonds for IrB7 and seven σ bonds for RuB8, three lone pairs on the Ir and Ru atom, and six completely delocalized σ and π bonds for both clusters. The AdNDP analysis of the umbrella-like AgB9 cluster gives more complicated bonding characters, that is, an extra 2c–2e σ bond located on Ag−B besides the seven 2c–2e σ bonds on the B−B bonds of circumference. Moreover, five lone pairs on the Ag atom and six delocalized 9c–2e σ and π bonds on the B9 ligand are revealed. The wheel-like IrB9 has nine 2c–2e σ B−B bonds, three lone pairs on the Ir atom, and six completely delocalized σ and π bonds, obeying the 4N + 2 Hückel rule of aromaticity and consistent with the results by Li et al. [21] In addition, the AdNDP analyses reveal that the clusters with high spin state have similar chemical bonding characteristic with the closed-shell ones in the same structural patterns. In other words, the chemical bonding characteristic is closely related to the structural pattern, while the spin state of cluster may only affect the specific occupations of some bonding orbitals by single or dual electrons (see Fig. S5 e–f).

The NICS(0)zz at the geometry center of half-sandwich clusters in the range of − 81.8 to − 105.3 ppm (Table 2) suggest their very strong aromatic character. The umbrella-like AgB9 and AuB9 clusters also have considerable NICS(0)zz values (− 61.4 and − 52.9 ppm). The NICSzz at a point 1.0 Å above the ring center (NICS(1)zz) of wheel-like IrB9 cluster is − 75.9 ppm, which is much larger than benzene (− 10.3 ppm) and corresponds to the doubly aromaticity. These values are comparable to the NICS(1)zz values of wheel-like CoB9 cluster (− 85.4 ppm) reported by Ito et al. [19] In other words, aromaticity or doubly aromaticity plays an essential role in the stability of these MBn clusters.

3.4 Two-dimensional superatomic clusters

From the above discussions, all these relatively stable clusters adopt nearly circular disk geometries with completely delocalized π system obeying the 4N + 2 Hückel rule of aromaticity. Intuitively, it is tempting to ascribe the enhanced stability of these MBn clusters to the closed electron shell in a disk jellium model [54, 55], which has been successfully applied to pure and transition metal-doped Au clusters as well as P- or Si-doped Bn clusters [56,57,58,59]. In the case of circular clusters with superatomic orbital in sequence of 1S21P41D42S21F42P41G42D43S2…, enhanced stabilities are expected to occur at 2, 6, 10, 12, 16, 20, 24, 28, 30… valence electrons, which are in fact related to the concept of “aromaticity” [60]. For instance, the aromaticity of a benzene molecule arises from six delocalized p electrons forming a delocalized π system in a hexagonal confinement, and the electronic configuration is thus analogous to 1S21P4.

The energy levels and spatial distributions of the occupied molecular orbitals of these relatively stable clusters (e.g., IrB7, RuB8, IrB9 and AgB9) are illustrated in Fig. 4. Obviously, all of them behave as 2D superatoms with orbitals that can be categorized into 2D superatomic molecular orbitals. Interestingly, three orbitals (with possible degeneracy) in each cluster are recognized as π bonds, which are consistent with the delocalized π-bonding based on the AdNDP analyses. For example, HOMO‒4 and HOMO‒6 of IrB7, HOMO‒3 and HOMO‒6 of RuB8, HOMO‒2 and HOMO‒5 of IrB9 as well as HOMO‒2 and HOMO‒9 of AgB9 cluster. Those π bonds are perpendicular to the circular disk and distributed over the upper and lower surfaces, which play an important role in the structural stability of circular disk. The valence electrons distributed in-plane are participating in the 2D superatomic orbitals except the lone pairs on M atom.

Fig. 4
figure 4

Energy levels and corresponding molecule orbitals of IrB7, RuB8, IrB9 and AgB9 clusters. The unoccupied orbital are indicated in dotted line. The arrows indicate the π bonding orbitals

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After subtracting the six π-bonding electrons, the half-sandwich IrB7 and RuB8 clusters have 24 and 26 remaining valence electrons, respectively, forming a closed electronic shell 1S21P41D42S21F4(1F2)2P42D43S2 configuration. The wheel-like IrB9 cluster has 30 remaining valence electrons: 2 electrons (\({\text{d}}_{{\text{z}}^{2}}\) lone pair) being localized on the Ir atom (HOMO‒1) and the rest 28 electrons forming a closed electronic shell 1S21P41D41F42S21G42D42P4 configuration. The umbrella-like AgB9 has 32 remaining valence electrons: 5 lone pairs on the Ag atom (HOMO‒4, HOMO‒6, HOMO‒7), one σ bond on Ag‒B (HOMO) and the rest 20 electrons forming a closed electronic shell 1S21P41D42S21F42P4 configuration.

4 Conclusions

A systematic investigation on the structural and electronic properties of 4d and 5d M-doped Bn (n = 7−10) clusters has been performed using the comprehensive genetic algorithm structural searches combined with DFT calculations. For both 4d and 5d transition metal elements, MBn clusters adopt hypercoordinated half-sandwich for early M dopants, wheel-like structures for middle M dopants and lower-coordinated quasi-planar or umbrella-like structures for late M dopants. Strikingly, the half-sandwich RhB7, IrB7, RuB8 and OsB8; wheel-like IrB9; and umbrella-like AgB9 and AuB9 clusters exhibit high stability, which carry “magic number” valence electrons occupying the closed electronic shells and following a disk jellium model. Chemical bonding analysis reveals that the rest delocalized π bonds distributed over the circular disk play a key role in their structural stability. These theoretical results establish a complete picture of the 4d and 5d M-doped boron clusters, which have high symmetry and appreciable stability. The present study may promote experimental synthesis of these intriguing B-based 2D superatom clusters as building block of nanomaterials.