Boron clusters with 46, 48, and 50 atoms: competition among the core–shell, bilayer and quasi-planar structures

Linwei Sai a, Xue Wu b, Nan Gao b, Jijun Zhao *b and R. Bruce King c
aDepartment of Mathematics and Physics, Hohai University, Changzhou 213022, China
bKey Laboratory of Materials Modification by Laser, Ion and Electron Beams (Dalian University of Technology), Ministry of Education, Dalian 116024, China. E-mail:
cDepartment of Chemistry and Center for Computational Chemistry, University of Georgia, Athens, GA 30602, USA

Received 5th April 2017 , Accepted 16th June 2017

First published on 20th June 2017

Using a genetic algorithm combined with density functional theory calculations, we perform a global search for the lowest-energy structures of Bn clusters with n = 46, 48, 50. Competition among different structural motifs including a hollow cage, core–shell, bilayer, and quasi-planar, is investigated. For B46, a core–shell B4@B42 structure resembling the larger Bn clusters with n ≥ 68 is found to compete with a quasi-planar structure with a central hexagonal hole. A quasi-planar configuration with two connected hexagonal holes is most favorable for B50. More interestingly, an unprecedented bilayer structure is unveiled at B48, which can be extended to a two-dimensional bilayer phase exhibiting appreciable stability. Our results suggest alternatives to the cage motif as lower-energy Bn cluster structures with n > 50.

The discovery of carbon fullerenes has stimulated considerable interest in the search for related cage clusters containing elements other than carbon.1 Boron, as the neighbor of carbon in the periodic table, also favors sp2 hybridization like carbon. Intuitively, boron cages, also known as borospherenes,2 exhibiting relatively high stability analogous to carbon fullerenes are expected to exist as viable species. During the past two decades, gas-phase boron clusters have been intensively investigated using experimental photoelectron spectroscopy combined with ab initio calculations.3–5

In 2007, Yakobson's group6 predicted a highly stable B80 cage of icosahedral symmetry exhibiting an appreciable HOMO–LUMO gap of about 1 eV. This B80 cage with 240 valence electrons is an isoelectronic and isostructural analogue of the famous C60 buckyball. However, ab initio global search by us7,8 and other groups9,10 revealed that more compact B12-centered core–shell structures (as the precursor of bulk boron solids) rather than hollow cages are energetically more favorable for the large Bn clusters with n ≥ 68, mainly owing to the electron-deficient nature of boron.

More recent studies have led to some breakthroughs in uncovering novel structures for boron cages in the intermediate size range around n = 40. A theoretical study combining photoelectron spectroscopy with ab initio global search by Zhai et al.2 identified a D2d cage of B40 exhibiting an extremely low electron binding energy. Meanwhile, an independent theoretical study by Lv et al.11 disclosed a D2h cage of B38 with a large HOMO–LUMO gap and high double aromaticity. Subsequent theoretical and experimental studies have led to a series of borospherenes at B29,12 B39,13 B41+,14 B42+,14,15 and B44.16 Using an ab initio global search, our group17 recently discovered the smallest all-boron cage at B28, shortly thereafter, our theoretical prediction was confirmed experimentally by Wang and coworkers.18

In addition to the boron cages, several other structural motifs are also found in the medium-sized Bn clusters for n = 20–56. These include double-ring or three-ring tubular configurations,19,20 bowl shapes with a central pentagonal or hexagonal hole,21–23 and quasi-planar structures containing holes in a triangular network.24–26 Clearly, the coexistence of these diverse structural patterns reflects the complexity in the potential energy surface (PES) of boron clusters, making this area of research very challenging.

Despite the progress outlined above, little is known about the most stable structures of relatively large Bn clusters beyond n = 46. Nevertheless, sufficiently large Bn clusters with n ≥ 68 have been shown to adopt compact core–shell structures. A question of interest is the minimum size Bn cluster favoring core–shell geometry. Clearly, this puzzle is a missing link in a complete picture of the structural evolution of boron clusters. In this Communication, we address this issue by performing ab initio global search of selected Bn clusters at n = 46, 48, 50. We find a delicate balance between energetically preferred structures in this size range. Thus the most preferred structural type changes drastically from core–shell (or quasi-planar) at B46 to bilayer at B48, and then quasi-planar at B50. This suggests strong competition between the core–shell and monolayer/bilayer structural motifs in large boron clusters and boron nanostructures.

The PES of Bn (n = 46, 48, 50) clusters were globally explored using our own comprehensive genetic algorithm (CGA) code incorporated in density functional theory (DFT) calculations. The details can be found in a recent review article.27 The validity and efficiency of this CGA-DFT scheme have been well demonstrated in our recent studies on Pt–Sn,28 V–Si,29,30 Si–B,31 and Ge–Fe32 clusters.

For each cluster size, we performed a few independent GA search programs with different presumed symmetries, including the point groups C2, C3, C5, and Cs. Within the symmetry constraint, each GA search lasted for 3000 iterations by retaining sixteen members in the population. At every iteration, the child cluster configuration generated from the crossover and mutation operations was fully relaxed with DFT optimization. The double numerical basis including d-polarization functions, and the Perdew–Burke–Enzerhof (PBE) functional33 within the generalized gradient approximation (GGA), as implemented in the DMol3 program,34 were adopted. Self-consistent field electronic structure calculations were done with a convergence criterion of 10−6 a.u. on the total energy.

In a previous benchmark study,8 we have demonstrated that the TPSSh functional35 combined with the 6-311G(d) basis set can reproduce the energy sequence of different isomers of boron clusters found in high-level CCSD(T) calculations. We therefore re-optimized those low-energy isomer structures from the CGA-DFT global search at the TPSSh/6-311G(d) level of theory using the Gaussian09 program.36 Vibrational analysis of Bn clusters in their equilibrium configurations at the TPSSh/6-311G(d) level of theory were carried out to ensure that there is no imaginary frequency corresponding to the saddle point on the PES and to include the zero-point-energy (ZPE).

The lowest-energy configurations and several representative isomers for Bn clusters (n = 46, 48, 50) from the CGA-DFT search are shown in Fig. 1 and their electronic properties are provided in Table 1. Many other isomers considered for these Bn clusters are presented in Fig. S1–S3 and Tables S1–S3 of the ESI. In the size range explored, there is strong competition among different structural motifs, i.e., hollow cage, quasi-planar, core–shell, bilayer. Even though the hollow cage is the dominant motif for Bn clusters in the size range n = 38–44,2,11,13–16 none of the cage isomers in Fig. 1 and Fig. S1–S3 is the ground state for Bn clusters with n = 46, 48, 50. The closest case is the C2v cage at B46, which is less stable than the lowest-energy isomer by 0.88 eV.

image file: c7nr02399e-f1.tif
Fig. 1 Representative structures of B46, B48 and B50 clusters belonging to the core–shell, cage, bilayer and quasi-planar motifs from TPSSh/6-311G(d) calculations. The interior atoms of the core–shell structures are highlighted in blue.
Table 1 Binding energy (Eb), HOMO–LUMO gap (EHL), and the coordination number (CN) of the representative isomers of Bn clusters (n = 46, 48, 50) with structures given in Fig. 1 from TPSSh/6-311G(d) calculations
Isomer Structure E b (eV) E HL (eV) CN
B46(I) Core–shell 5.07 1.56 5.26
B46(II) Quasi-planar 5.07 0.80 4.78
B46(III) Bilayer 5.06 1.37 5.00
B46(IV) Cage 5.05 1.09 4.78
B48(I) Bilayer 5.10 0.93 5.67
B48(II) Core–shell 5.07 1.09 5.71
B48(III) Quasi-planar 5.06 1.15 4.67
B48(IV) Cage 5.06 0.96 4.71
B50(I) Quasi-planar 5.10 1.28 4.76
B50(II) Core–shell 5.08 0.88 5.52
B50(III) Cage 5.07 0.89 4.88
B50(IV) Bilayer 5.06 0.91 4.72

For a given size, the relative stability of cluster isomers may rely on the description of the exchange–correlation interaction and the inclusion of the temperature effect. To further confirm the results by TPSSh calculations, we considered a few isomers of B20 and compared their relative energies using different methods, including CCSD(T), MP2, B2PLYP,37 HSE06,38 and TPSSh. The results are summarized in Table S4 of the ESI. One can see that MP2 and B2PLYP methods cannot reproduce the correct energy order of isomers by CCSD(T) calculation. Both TPSSh and HSE06 are able to reasonably give the energy difference between the different isomers by CCSD(T), while TPSSh performs slightly better. Then, we used HSE06 to recalculate the important isomers in Fig. 1 to examine the dependence of the energy sequence on the choice of the exchange–correlation functional. The temperature correction for all isomers in Fig. 1 was also included using both TPSSh and HSE06 methods combined with the 6-311G(d) basis set. The results at 0 K and 298 K are presented in Table S5 of the ESI.

Surprisingly, a core–shell structure with four endohedral atoms inside a 42-atom cage is most favorable for B46 from zero-temperature TPSSh calculation. The outer cage in the B4@B42 core–shell structure for B46 has 42 vertices, 100 edges, and 60 faces including 52 triangles, 4 pentagons, and 4 hexagons. Among the 42 exterior boron atoms, four groups of seven atoms are clustered into concave B@B6 umbrellas39 with the indented central atom of each umbrella bonded to a boron atom of the interstitial B4 unit. Despite the complete lack of symmetry of this 42-vertex polyhedron, it can be considered as a highly distorted version of a 42-vertex deltahedron of icosahedral symmetry that can be obtained from the face centers of the C80 fullerene deltahedron.40 This distortion, including the effect of the interstitial B4 unit, breaks edges in triplets of adjacent triangular faces to generate pentagonal faces and in quartets of adjacent triangular faces to generate hexagonal faces.

To the best of our knowledge, this is the smallest core–shell structure found for boron clusters. The onset of the core–shell structure in B46 with a large HOMO–LUMO gap of 1.56 eV can be associated with the dominant core–shell structures for Bn clusters with n ≥ 68,7 which is a consequence of the electron deficiency of boron. The average coordination number (CN) of the B4@B42 structure is 5.26, which is larger than the other three isomers shown in Fig. 1 and only slightly smaller than CN = 5.45 for the B12@B68 core–shell structure of B80.

At room temperature (298 K), however, the core–shell configuration becomes less favorable than the quasi-planar one by 0.11 eV from TPSSh calculation (see Table S5). Meanwhile, the HSE06 functional always favors the quasi-planar structure with energy differences of 0.05 eV at 0 K and 0.36 eV at 298 K, respectively. As depicted in Table S5, this is the only case where the choice of the functional and the inclusion of the temperature effect may change the lowest-energy structure of a boron cluster. This also suggests strong competition between these two structural motifs for B46.

Our most noteworthy finding is the previously unreported lowest-energy bilayer structure for B48. As shown in Fig. 2, this D2h bilayer structure consists of two identical quasi-planar layers of nineteen atoms, bridged by a B5 chain at each end. As the major building unit of bilayer B48, the upper or lower quasi-planar B19 structure can be derived from a piece of a triangle lattice with C6v symmetry. Six of the peripheral atoms and two of the inner-circle atoms of such a lattice can then undergo large out-of-plane buckling to form B–B bonds with the bridge atoms and the atoms in another B19 layer, respectively. A similar bilayer structural motif also exists for the other sized Bn clusters, but is energetically less favorable, i.e., ΔE = 0.68 eV for B46 and ΔE = 2.00 eV for B50 relative to the corresponding ground states.

image file: c7nr02399e-f2.tif
Fig. 2 (a) Top view and (b) side view of the bilayer B48 cluster, (c) top view and (d) side view of the bilayer boron sheet constructed following the B48 motif. The out-of-plane atoms are highlighted in blue. Red lines in (a) and (c) highlight the shared structural unit in both cluster and bilayer sheet. Dashed lines in (c) show the unit cell.

Starting from the bilayer structure of B48, an extended two-dimensional (2D) bilayer allotrope of boron can be constructed (Fig. 2). This novel 2D structure is more stable than a recently reported bilayer boron sheet with the P6/mmm space group41 by 0.10 eV per atom and the monolayer α-boron sheet42 by 0.16 eV per atom, respectively. In this regard, the bilayer structure at B48 is a precursor of the 2D bilayer boron allotrope. As displayed in Fig. S4 of the ESI, this newly found 2D phase of boron is dynamically stable with no imaginary phonon branch and metallic with a few dispersive bands crossing the Fermi level.

Unexpectedly, the lowest-energy configuration of B50 is neither a core–shell nor a bilayer. Instead a quasi-planar structure with a pair of connected hexagonal holes is found (Fig. 1). This structure has been recently reported by Xu et al.25 In fact, the double hexagonal vacancy was first discovered in B35 (ref. 24) and recently found in B37 and B38,43 and it was also featured in the co-existing isomers of B40.2 The connected hexagonal holes also exist in the monolayer β-boron sheet.42 The energy difference between the quasi-planar and bilayer B50 structures is more substantial at 2.00 eV.

A core–shell B@B49 structure lies 0.94 eV in energy above the most stable quasiplanar B50 structure (Fig. 1). This is the first example of a boron cluster structure with a single endohedral boron atom inside a boron cage. The outer B49 polyhedron of this structure has 49 vertices, 133 edges, and 86 faces. Four of the faces are pentagons and the remaining 82 faces are triangles. The surface of this B49 polyhedron has five concave B@B6 hexagonal umbrellas and six convex B@B5 pentagonal umbrellas. The inward oriented central vertices of the five concave B@B6 umbrellas are bonded to the interstitial boron atom leading to tetragonal pyramidal coordination.

Among all the isomers considered here, we find that the bilayer B48 and quasiplanar B50 possess the largest binding energies (5.10 eV per atom by TPSSh calculation, 5.33 and 5.32 eV per atom by HSE06 calculation, also see Table S6). Their higher stabilities can be related to the aromaticity arising from electron delocalization in complete circuits.44 Large negative values of the nucleus independent chemical shift (NICS) are obtained, i.e., −30.4 ppm for bilayer B48 and −39.6 ppm for quasiplanar B50, respectively. These values are comparable to previously reported NICS values for smaller boron clusters such as B28[thin space (1/6-em)]17 and B38.11 Accordingly, a pronounced feature of a symmetrically distributed π bond can be seen in the HOMO and LUMO23 of these two clusters (Fig. 3), as compared with those of other cluster isomers shown in Fig. S5 of the ESI.

image file: c7nr02399e-f3.tif
Fig. 3 The highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO): (a) bilayer B48, (b) quasi planar B50.

To summarize, our ab initio global search has revealed a rich variety of unprecedented structures for medium-sized boron clusters having 46, 48, and 50 atoms. We find co-existence and competition among three major structural motifs, namely core–shell, bilayer, and quasi planar structures. Hollow cages are also found as metastable isomers for the Bn (n = 46, 48, 50) systems. Inspired by the lowest-energy structure of B48, a new bilayer phase of a 2D boron allotrope with high stability is found. Our findings not only represent a missing link between the cage and core–shell motifs during the structural evolution of Bn clusters, but also provide some key insights into understanding the structure and bonding of various boron nanostructures.


This work was supported by the National Natural Science Foundation of China (11574040), the Fundamental Research Funds for the Central Universities (DUT16-LAB01, DUT17LAB19, 2014B16514), and the Supercomputing Center of Dalian University of Technology.


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Electronic supplementary information (ESI) available: Structures and energetic data of the low-lying isomers of B46, B48 and B50 clusters. Benchmark calculations on B20. The HOMO and LUMO of B46, B48 and B50. Relative energies by TPSSh and HSE06 calculations at 0 K and 298 K. Phonon dispersion and the band structure of a bilayer boron sheet. Coordinate files of the representative structures of B46, B48 and B50. See DOI: 10.1039/c7nr02399e
These authors contributed equally to this work.

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