B63: The Most Stable Bilayer Structure with Dual Aromaticity.

The emergence of a bilayer B48 cluster, which has been both theoretically predicted and experimentally observed, as well as the recent experimental synthesis of bilayer borophene sheets on Ag and Cu surfaces, has generated tremendous curiosity in the bilayer structures of boron clusters. However, the connection between bilayer boron cluster and bilayer borophene remains unknown. By combining a genetic algorithm and density functional theory calculations, a global search for the low-energy structures of the B63 cluster was conducted, revealing that the Cs bilayer structure with three interlayer B-B bonds is the most stable bilayer structure. This structure was further examined in terms of its structural stability, chemical bonding, and aromaticity. Interestingly, the interlayer bonds induce strong electronegativity and robust aromaticity. Furthermore, the dual aromaticity stems from diatropic currents originating from virtual translational transitions for both σ and π electrons. This unprecedent bilayer boron cluster is anticipated to enrich the concept of dual aromaticity and serve as a potential precursor for bilayer borophene.

Predicting Binding Energies and Electronic Properties of Boron Nitride Fullerenes Using a Graph Convolutional Network.

As isoelectronic counterparts of carbon fullerenes, medium-sized boron nitride clusters also prefer cage structures composed of even-sized polygons. As the cluster size increases, the number of cage isomers grows rapidly, and determining the ground state structure requires a tremendous amount of DFT calculations. Herein, we develop a graph convolutional network (GCN) that can describe the energy of a (BN)n cage by its topology connection. We define a vertex feature vector on a dual polyhedron by the permutation of the neighbor vertices' degree and aggregate the information on vertices by two graph convolutional layers to learn the local feature of the dual polyhedron. The GCN is trained on (BN)28 and subsequently tested on (BN)23 and (BN)24 data sets, which satisfactorily reproduce the order of isomer energies from DFT calculations. We further employ the trained GCN to predict the ground state structures within the size range of n = 25-32, which agree well with DFT results. Using the same GCN framework, we also successfully trained the highest-occupied or lowest-unoccupied orbital energies of (BN)28 isomers. The present graph convolutional network establishes a direct mapping between the topological connection and the energetic or electronic properties of a cage-like cluster or molecule.

Pivotal role of the B12-core in the structural evolution of B52-B64 clusters.

An icosahedral B12 cage is a basic building block of various boron allotropes, and it also plays a vital role in augmenting the stability of fullerene-like boron nanoclusters. However, the evolution of compact core-shell structures is still a puzzle. Using a genetic algorithm combined with density functional theory calculations, we have performed a global search for the lowest-energy structures of Bn clusters with n = 52-64, which reveals that bilayer and core-shell motifs frequently alternate as the ground state. Their structural stability is assessed, and the competition mechanism between various patterns is also elucidated. More interestingly, an unprecedented icosahedral B12-core half-covered structure is identified at B58, which bridges the gap between the smallest core-shell B4@B42 and the complete core-shell B12@B84 cluster. Our findings provide valuable insights into the bonding pattern and growth behavior of medium-sized boron clusters, which facilitate the experimental synthesis of boron nanostructures.

Unprecedented Prediction of a B160 Cluster Stuffed by Dual-Icosahedron B12.

Serving as the premise to understand bulk allotropes, boron clusters have been intriguing experimentalists and theoreticians to study their geometries and chemical bonding. Here, we designed a complete core-shell B160 cluster stuffed by two B12 cores, which is energetically preferable over the bilayer structure of the same size. The unprecedented peanutlike structure with Ci symmetry has superior stability and exhibits superatomic electronic configuration and spherical aromaticity. Our theoretical work not only proposed the core-shell structure of dual icosahedrons for the first time but also indicated the multi-B12 core-shell structural pattern in boron particles, bridging to boron crystalline structures.

B96: a complete core-shell structure with high symmetry.

A complete core-shell and highly symmetric B96 was designed. The core-shell B96 of Th symmetry is energetically favorable compared to the bilayer motif and the core-shell structure can be well maintained during first-principles molecular dynamics simulations at high temperatures (up to 1000 K). Moreover, it exhibits a superatomic electronic configuration and spherical aromaticity. Our theoretical work not only confirmed that the core-shell structural pattern is more energetically favorable for large-sized boron clusters, but also provided a strategy to design large boron clusters with a core-shell structure of high symmetry.

Artificial neural network potential for Au20clusters based on the first-principles.

The search of ground-state structures (GSSs) of gold (Au) clusters is a formidable challenge due to the complexity of potential energy surface (PES). In this work, we have built a high-dimensional artificial neural network (ANN) potential to describe the PES of Au20clusters. The ANN potential is trained through learning the GSS search process of Au20by the combination of density functional theory (DFT) method and genetic algorithm. The root mean square errors of energy and force are 7.72 meV atom-1and 217.02 meV Å-1, respectively. As a result, it can find the lowest-energy structure (LES) of Au20clusters that is consistent with previous results. Furthermore, the scalability test shows that it can predict the energy of smaller size Au16-19clusters with errors less than 22.85 meV atom-1, and for larger size Au21-25clusters, the errors are below 36.94 meV atom-1. Extra attention should be paid to its accuracy for Au21-25clusters. Applying the ANN to search the GSSs of Au16-25, we discover two new structures of Au16and Au21that are not reported before and several candidate LESs of Au16-18. In summary, this work proves that an ANN potential trained for specific size clusters could reproduce the GSS search process by DFT and be applied in the GSS search of smaller size clusters nearby. Therefore, we claim that building ANN potential based on DFT data is one of the most promising ways to effectively accelerate the GSS pre-screening of clusters.

Theoretical Design of Novel Boron-Based Nanowires via Inverse Sandwich Clusters.

Borophene has important application value, boron nanomaterials doped with transition metal have wondrous structures and chemical bonding. However, little attention was paid to the boron nanowires (NWs). Inspired by the novel metal boron clusters Ln2B n - (Ln = La, Pr, Tb, n = 7-9) adopting inverse sandwich configuration, we examined Sc2B8 and Y2B8 clusters in such novel structure and found that they are the global minima and show good stability. Thus, based on the novel structural moiety and first-principles calculations, we connected the inverse sandwich clusters into one-dimensional (1D) nanowires by sharing B-B bridges between adjacent clusters, and the 1D-Sc4B24 and 1D-Y2B12 were reached after structural relaxation. The two nanowires were identified to be stable in thermodynamical, dynamical and thermal aspects. Both nanowires are nonmagnetic, the 1D-Sc4B24 NW is a direct-bandgap semiconductor, while the 1D-Y2B12 NW shows metallic feature. Our theoretical results revealed that the inverse sandwich structure is the most energy-favored configuration for transition metal borides Sc2B8 and Y2B8, and the inverse sandwich motif can be extended to 1D nanowires, providing useful guidance for designing novel boron-based nanowires with diverse electronic properties.

Ground-State Structures of Hydrated Calcium Ion Clusters From Comprehensive Genetic Algorithm Search.

We searched the lowest-energy structures of hydrated calcium ion clusters Ca2+(H2O)n (n = 10-18) in the whole potential energy surface by the comprehensive genetic algorithm (CGA). The lowest-energy structures of Ca2+(H2O)10-12 clusters show that Ca2+ is always surrounded by six H2O molecules in the first shell. The number of first-shell water molecules changes from six to eight at n = 12. In the range of n = 12-18, the number of first-shell water molecules fluctuates between seven and eight, meaning that the cluster could pack the water molecules in the outer shell even though the inner shell is not full. Meanwhile, the number of water molecules in the second shell and the total hydrogen bonds increase with an increase in the cluster size. The distance between Ca2+ and the adjacent water molecules increases, while the average adjacent O-O distance decreases as the cluster size increases, indicating that the interaction between Ca2+ and the adjacent water molecules becomes weaker and the interaction between water molecules becomes stronger. The interaction energy and natural bond orbital results show that the interaction between Ca2+ and the water molecules is mainly derived from the interaction between Ca2+ and the adjacent water molecules. The charge transfer from the lone pair electron orbital of adjacent oxygen atoms to the empty orbital of Ca2+ plays a leading role in the interaction between Ca2+ and water molecules.

Competition between tubular, planar and cage geometries: a complete picture of structural evolution of Bn (n = 31-50) clusters.

Stimulated by the early theoretical prediction of B80 fullerene and the experimental finding of the B40 cage, the structures of medium-sized boron clusters have attracted intensive research interest during the last decade, but a complete picture of their size-dependent structural evolution remains a puzzle. Using a genetic algorithm combined with density-functional theory calculations, we have performed a systematic global search for the low-lying structures of neutral Bn clusters with n = 31-50. Diverse structural patterns, including tubular, quasi-planar, cage, core-shell, and bilayer, are demonstrated for the ground-state Bn clusters; for certain cluster sizes, unprecedented geometries are predicted for the first time. Their stabilities at finite temperatures are evaluated, and the competition mechanism between various patterns is elucidated. Chemical bonding analysis reveals that the availability of localized σ bonds and delocalized π bonds in the Bn clusters play a key role in their structural stability. Our results provide important insights into the bonding pattern and growth behavior of medium-sized boron clusters, which lay the foundation for experimental design and synthesis of boron nanostructures.

A super stable assembled P nanowire with variant structural and magnetic/electronic properties via transition metal adsorption.

By means of first-principles calculations, we systematically investigated the structure, stability and magnetic and electronic properties of one-dimensional P nanowire (1D-P10 NW) assembled by Pn subunits (n = 2, 8) and transition metal doped 1D-P10 NW. Our calculations showed that the assembled 1D-P10 NW is super stable in thermodynamic, dynamic, thermal and chemical perspectives. Moreover, when the assembled 1D-P10 NW is decorated with transition metals (TM = Ti ∼ Zn, Zr ∼ Mo), structural transformation occurs (to sandwich or quasi-sandwich chains), and various magnetic and electronic characteristics are introduced to the nanowire. Particularly, the sandwich chains 1D-Mn2@P10 and 1D-V1@P5 are a ferromagnetic semiconductor and a ferromagnetic half-metal, respectively, and the magnetic anisotropy energies are both ∼0.3 meV per Mn/V atom. Our theoretical studies proposed a super stable 1D P nanowire and also offer a feasible approach to reach P5-TM-P5-TM chains with diverse magnetic and electronic properties, as well as ferromagnetic vdW-type 2D systems, which are promising in nanoelectronic devices and spintronics.

Structural and Electronic Properties of Binary Clusters SimGen (m + n = 6-13).

Using genetic algorithm combined with density functional theory calculations, we performed an unbiased global search for the most-stable structures of binary clusters SimGen with size s = m + n from 6 to 13. Further, we studied the structural and electronic properties of SimGen clusters using the B3LYP and CCSD(T) methods coupled with 6-311G + (d) basis set. For s = 6-12, SimGes-m clusters exhibit similar geometries to Sis and Ges clusters, respectively. However, for s = 13, the geometries of SimGes-m clusters fall into five completely different patterns. The negative mixing energies of SimGes-m clusters indicate that they possess higher energetic stability than Sis and Ges clusters. Among all clusters investigated, Si2Ge4, Si2Ge5, Si2Ge6, Si6Ge3, Si5Ge5, Si7Ge4, Si3Ge9, and Si8Ge5 clusters have the relatively lower mixing energies and thus the highest energetic stabilities. Moreover, the Si2Ge4, Si2Ge6, Si5Ge5, Si7Ge4, and Si8Ge5 clusters with higher HOMO-LOMO gaps should have higher chemical stabilities than the same-sized Sis and Ges clusters. The Si5Ge5 cluster has a higher ionization potential than Si10 and Ge10. At the size s = 13, the geometry with the highest symmetry has the highest energetic and chemical stabilities.

Evolution of atomic structures of SnN, SnN -, and SnNCl- clusters (N = 4-20): Insight from ab initio calculations.

An unbiased global search was employed to explore the low-energy structures of SnN, SnN -, and SnNCl- clusters with N = 4-20 atoms based on the genetic algorithm combined with density functional theory calculations. Some unprecedented low-energy isomers are reported for SnN and SnNCl- clusters. The theoretical electronic properties such as binding energy per atom, ionization potential, adiabatic detachment energy, and vertical detachment energy compare well with the experimental data. Based on the equilibrium structures, the simulated photoelectron spectra are in good agreement with the experimental data in the range of N = 4-20. With addition of a Cl atom on the SnN - cluster, which causes almost no rearrangement on the structural framework, the first peaks in all original photoelectron spectra of SnN - clusters disappear and other peaks nearly retain the original feature at most sizes.

Revisit the landscape of protonated water clusters H+(H2O)n with n = 10-17: An ab initio global search.

Using a genetic algorithm incorporated with density functional theory, we explore the ground state structures of protonated water clusters H+(H2O)n with n = 10-17. Then we re-optimize the isomers at B97-D/aug-cc-pVDZ level of theory. The extra proton connects with a H2O molecule to form a H3O+ ion in all H+(H2O)10-17 clusters. The lowest-energy structures adopt a monocage form at n = 10-16 and core-shell structure at n = 17 based on the MP2/aug-cc-pVTZ//B97-D/aug-cc-pVDZ+ZPE single-point-energy calculation. Using second-order vibrational perturbation theory, we further calculate the infrared spectra with anharmonic correction for the ground state structures of H+(H2O)10-17 clusters at the PBE0/aug-cc-pVDZ level. The anharmonic correction to the spectra is crucial since it reproduces the experimental results quite well. The extra proton weakens the O-H bond strength in the H3O+ ion since the Wiberg bond order of the O-H bond in the H3O+ ion is smaller than that in H2O molecules, which causes a red shift of the O-H stretching mode in the H3O+ ion.

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

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.

Structural evolution and electronic properties of medium-sized gallium clusters from ab initio genetic algorithm search.

Using genetic algorithm incorporated with density functional theory, we have explored the size evolution of structural and electronic properties of neutral gallium clusters of 20-40 atoms in terms of their ground state structures, binding energies, second differences of energy, HOMO-LUMO gaps, distributions of bond length and bond angle, and electron density of states. In the size range studied, the Ga(n) clusters exhibit several growth patterns, and the core-shell structures become dominant from Ga31. With high point group symmetries, Ga23 and Ga36 show particularly high stability and Ga36 owns a large HOMO-LUMO gap. The atomic structures and electronic states of Ga(n) clusters significantly differ from the a solid but resemble beta solid and liquid to certain extent.

A topological method for global optimization of clusters: application to (TiO2)n (n = 1-6).

A new topological method is presented to generate the isomer structures of compound clusters with well defined covalent bonds. This method, combined with density functional theory, has been used to perform global optimization of (TiO(2))(n) (n = 1-6) clusters. Our comprehensive search not only reproduces all of the known lowest-energy structures reported in previous works but also reveals some new low-energy structures. Some energetically unfavorable motifs that induce energy penalties are obtained and discussed. Based on the ground state structures of the anionic (TiO(2))(n). clusters, the electron affinities and photoelectron spectra are simulated and compared with available experimental data.

Lowest-energy structures and electronic properties of Na-Si binary clusters from ab initio global search.

The ground state structures of neutral and anionic clusters of Na(n)Si(m) (1 ≤ n ≤ 3, 1 ≤ m ≤ 11) have been determined using genetic algorithm incorporated in first principles total energy code. The size dependence of the structural and electronic properties is discussed in detail. It is found that the lowest-energy structures of Na(n)Si(m) clusters resemble those of the pure Si clusters. Interestingly, Na atoms in neutral Na(n)Si(m) clusters are usually well separated by the Si(m) skeleton, whereas Na atoms can form Na-Na bonds in some anionic clusters. The ionization potentials, adiabatic electron affinities, and photoelectron spectra are also calculated and the results compare well with the experimental data.