Structural disassembly, dangling bond and hydrogen storage of NinSn (n=2-10) clusters

Structural disassembly, dangling bond and hydrogen storage of NinSn (n=2-10) clusters | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Structural disassembly, dangling bond and hydrogen storage of Ni n S n ( n =2-10) clusters Weide Liu, Fenfa Zhang, Bowei Jin, Xuezhou Lu, Zhengyang Ding, and 2 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-5906730/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Computational cluster sciences are rooted in geometrical optimization successes of small groups of atoms or molecules. Following size growth and element increases, optimizations are arduous and hardly generalized in certain patterns, despite advances in calculation algorithm and computing powers. Herein, a disassembly-assembly strategy is introduced to reach stable structures of binary Ni n S n ( n = 2-10) clusters. The lowest-energy Ni n S n isomers can be viewed as nestifications of low-lying Ni n and S n components. Identical spatial orientations in and out of the Ni-S binary systems are kept for the elemental S n and Ni n clusters. At the same corresponding number n , the metal and chalcogenide clusters coincidentally possess the same point group and such a trend is kept up to the NiS crystalline phase. As a result, the growth pattern of the compound clusters is simply viewed as an assembly process of their component clusters. Furthermore, molecular orbital analysis indicates that the nested structures are endowed with a large amount of dangling bonds to adsorb hydrogen atoms for hydrogen storage. Such an assembly-disassembly route in structural optimizations is hoped to serve as a route map in future functional cluster predictions. Cluster structure hydrogen storage DFT investigation Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 1. INTRODUCTION Envisaged as an intermediate between molecules to bulk solids, clusters usually exhibit distinct structures, physical and chemical characteristics from their atomic and condensed counterparts[ 1 – 5 ]. A critical concern in understanding these quasi-particle properties stems from geometrical determinations of the most stable isomers within the sub-nanometer to nanometer scales. Different from the isolated atoms or bulk phases, spatial arrangements of the clusters seldom follow certain patterns that can be generalized for cluster peers. Despite the latest progress in algorithms and computational powers, typical optimization procedures still undergo isomer constructions, enumerations, energy comparison, and stability investigations. Yet, structural evolution of these “building bricks” has remained a mystery since the emergence of cluster science from the 1980s[ 6 – 9 ]. Geometric determinations are even more complicated in clusters containing few elements than the mono-element ones. Besides the aggregations of individual elements, the bonding between the elements brings additional factors during overall structural optimizations. A typical example can be retrieved from the transition-metal sulfide (TMS) clusters. Due to the bulks’ importance in catalysis, combustion, and biological systems[ 10 – 14 ], the TMS clusters have been extensively studied experimentally[ 15 – 24 ] and theoretically[ 25 – 30 ]. Moreover, the chalcogenides of simple stoichiometries, such as 1:1 and 1:2, are the most common and important TMS formula. Even though the TM x S y (TM = Mn, Fe, Co, Zn, and Cu, x = 1, y = 1 or 2) clusters have been well explored[ 26 – 31 ], the individual researches could not end up with a structural evolution theorem despite composition, stability, structure, and photodissociation elucidations. The composed TM x S y clusters do not abide by any geometrical patterns of the TM x or S y clusters. Structural connections between the compound and its component clusters may be faint or unexplored. However, it has been noticed that many macro-clusters can be divided by motifs from individual components, and the metal cluster cores are covered by organic shells[ 32 – 34 ]. Thus, it is important to check if such a dissembling scheme also exists in small clusters such as these TMSs. Hydrogen is an efficient, clean, and cost-effective energy source due to its high energy density. In recent years, H₂ has gained significant attention as a promising alternative energy carrier. Meanwhile, low-cost transition metal-based nanostructures with various dimensions and morphologies (e.g., 0D, 1D, 2D, and 3D), such as transition metal chalcogenides, have been extensively explored for energy applications[ 35 ]. Among them, MoS₂ has been widely studied as a promising catalyst for the hydrogen evolution reaction (HER) due to its unique electronic structure and high catalytic activity at edge sites[ 36 – 39 ]. Previous studies have shown that the incorporation of different elements can effectively modulate the catalytic properties of MoS₂, further enhancing its HER performance[ 40 – 43 ]. The Ni x S y binary system is ideal for sorting out the shortcuts in growth patterns of the compound clusters. The NiS is dimorphic with a hexagonal phase α-NiS at high temperature[ 44 – 47 ] and trigonal form β-NiS (millerite) at low temperature[ 48 – 54 ]. Both positive and negative Ni x S y clusters were verified experimentally through the laser ablation of NiS isotope-labeled samples[ 15 , 55 ]. The negative particles are constituted of clusters with similar fractions of Ni and S atoms. Besides, Ni and S formed clusters with a larger content of S in NiS n ( n = 1–4)[ 30 ] and preferably bonded at squeezed spaces[ 56 ]. The NiS also acted as a nanocrystal catalyst[ 57 ] or shielding component to cage clusters for efficient hydrogen evolution under different circumstances[ 58 ]. In this paper, we present a systematic investigation of Ni n S n ( n = 2–10) cluster properties, growth patterns, and their structural correlations to elementary Ni and S clusters by using density functional theory (DFT) calculations and particle swarm optimization. The calculated results demonstrate that Ni n S n clusters can be anatomized to Ni n and S n clusters with core/shell self-similarity morphology, which can be extended to NiS millerite and TMS crystal structure. The predicted NiS clusters own a large number of dangling bonds that facilitate the absorbance of hydrogen. This work provides a possible shortcut in structural optimizations for functional clusters by assembling or dissembling the compound clusters. 2. COMPUTATIONAL DETAILS Initial structures of Ni n S n clusters were generated using the CALYPSO software which has been verified highly efficient for cluster structures prediction[ 59 , 60 ]. A local version of the particle swarm optimization (PSO) algorithm was implemented to explore the potential energy surface for non-periodic systems[ 61 ]. In this work, Gaussian09 code[ 62 ] with PBEPBE/LANL2DZ[ 63 , 64 ] basis set was employed in local geometry optimization and total energy calculations. More than 2000 distinct isomers were selected from the CALYPSO results for finer optimization by using DFT implemented in the DMOL 3 program[ 65 ]. All-electron spin-unrestricted calculations were performed with a double numerical precision plus polarization with the addition of diffuse functions (DNP+). Relativistic calculations were carried out with scalar relativistic corrections to valence orbitals relevant to atomic bonding properties via a local pseudopotential with the Perdew -Burke-Ernzerhof (PBE) functional[ 66 ]. Convergence of self-consistent field (SCF) was set with a criterion of 1×10 − 5 Hartree on total energy and electron density, 2×10 − 3 Hartree/Å on the gradient, and 5×10 − 3 Å on the displacement, respectively. Harmonic vibrational analysis was carried out at the same level of theory to examine low-energy isomer stabilities and estimate zero-point energy corrections. To verify computational reliability, we performed a benchmark calculation on NiS dimer using different type of the gradient-corrected hybrid BP[ 67 ], BLYP[ 68 ], PW91[ 69 ] and PBE functionals in DMOL 3 for bond length and vibrational frequency. The results are listed in Table S1 , where the PBE method shows the best agreement with experimental values[ 70 – 72 ] and hence we selected the functional PBE in this study. 3. RESULTS AND DISCUSSION 3.1. Individual Ni n S n Cluster Study Various low-lying isomers of Ni n S n clusters ( n =1-10) are determined with global structure search and DFT calculations. The ground-state structures of Ni n S n isomers ( n up to 10) are presented in Figure 1. Other metastable isomers are listed in Figure S1 and Table S2 in the Supporting Information, in the order of relative energies and structural coordinates. Schematic representation in Figure 1 illustrates the anatomy of the lowest-energy structures of Ni n S n clusters. The diatomic NiS has a C ∞ v symmetry and a bond length of 1.986 Å. The planar rhombus with Ni-S bond length of 2.065 Å is the ground-state geometry for Ni 2 S 2 cluster that can be seen as a crisscross of Ni 2 and S 2 dimers. The Ni 3 S 3 has a quasi-planar equilateral triangle structure with a planar Ni 3 triangle coated by a S 3 triangle. Each bridging S atom is bonded with two Ni atoms in the cluster. The dimensional enhancement from 2D → 3D starts at n = 4 as shown in Figure 1. The square antiprism structure is the most stable structure for Ni 4 S 4 . The Ni 4 tetrahedron is embedded in the S 4 tetrahedron and each S atom is bonded to three Ni atoms through a face-capped model. Ni 5 S 5 cluster can be regarded as docking the square pyramids of Ni 5 and S 5 cluster with four Ni 3 S faces capping triangular. The most stable structure of Ni 6 S 6 can be generated by stacking the two Ni 3 S 3 triangles. Both the core Ni 6 and shell S 6 clusters are octahedrons with each S atom face-capping the Ni 3 triangle. From the cluster size of n =7, the Ni n S n cluster can be evolved by successively capping the N 6 S 6 . As shown later, the structures of Ni n and S n ( n =7-10) can be considered as mono-face-capped, bi-capped, tri-capped, and tetra-capped octahedral structures. A close look towards cluster geometries denotes 1) the compound clusters own self-similarity and nested-assembly characteristics to their component peers, and 2) a building-brick of an S atom face-capping a triangular Ni 3 turns out in all 3D clusters from n =4 to n =10. The former behavior indicates that the geometries and structural evolution of Ni n S n clusters may be relevant to these of the mono-compositional clusters, and the latter shows Ni 3 S tetrahedral configuration may act as a seed to drive cluster development. 3.2. Cluster Disassembly and Assembly The possibility of an assembly-disassembly scheme in the Ni n S n clusters is studied. Two important features towards the geometries of the Ni n S n clusters are noticed in Figure 1. (1) All the Ni n S n clusters have core-shell structures where the nickel stays as core and sulfur seats on shells. Nickel atoms stacked together, nucleated, and then subsequently bonded to sulfur atoms. The resulting Ni n S n clusters were stretched to three-dimensional (3D) composites when n ≥4. Each S atom favors face-capping on the triangular Ni plane via bonding with three Ni atoms in all 3D structures. On the contrary, distinct segregation between the sulfur atoms is observed due to the absence of S-S bonds. (2) The Ni n S n clusters can be viewed as assemblies of individual Ni n and S n clusters whose atoms keep the same spatial orientations as if they were in the binary systems. One “matryoshka doll” Ni n S n cluster owns a specifically oriented sulfur shell, which is filled by its nickel miniature. In another word, peeling away the S atoms emerges the Ni n cluster whose structure is similar to the S n cluster. To assemble the Ni n S n , the low-lying S n and Ni n clusters turn certain angles against and then inlay with each other to maximize spatial occupancies. In most cases, sulfur clusters are inversely placed to nickel ones as if the formers are turned 180 degrees or vice versa. Exceptions were found in Ni 2 S 2 and Ni 8 S 8 clusters, where 90 and 45 degrees clockwise rotations are applied respectively. Ni 7 S 7 cluster has a chiral structure as shown in Figure 1, which was predicted in bimetallic and trimetallic alloy clusters by doping hetero metal atoms to lower cluster symmetries[73,74]. Based on the above knowledge, Ni n S n are nested assemblies from the individual metallic and nonmetallic clusters. The individual Ni n and S n clusters have the same point group at a corresponding number n . The core Ni n cluster turns to the shell S n clusters when Ni atoms are replaced with sulfur atoms. Sulfur shell undergoes a self-similarity process from the Ni core. These small clusters can be viewed as inorganic self-similarity cases in the formation of geometric structures similar to the well-known double helix of DNA, which has not been reported in TMS clusters of MnS, FeS, and CoS [18,75]. We also note that the assembly-disassembly mechanism has been reported for single element clusters. Experimentally the supported Ag nanoclusters followed a construction pattern that can be separated for individual Ag clusters[76]. Similar behavior has also been observed in Si clusters. Previous DFT calculation and global minimum search have shown that the low-lying isomers of Si clusters can be derived from assembling tricapped trigonal prisms [77]. Thanks to the assembly-disassembly scheme, investigation of the growth patterns of the compound Ni n S n cluster is simplified to study its component Ni n or S n cluster. Here we take the Ni n cluster as an example and visualize its growth pattern in Figure 2. A successive Ni n+ 1 cluster can be generally evolved from a Ni n cluster by capping a Ni atom. Ni 6 is a 3D octahedral structure which is the basic building block for the core Ni n cluster from the size of n =6. Ni n ( n =7-10) clusters and can be viewed as mono-, bi-, tri- and tetra-capped octahedral structures. The peeled Ni n core clusters from the Ni n S n system obtained in this work differ from those of small bare nickel clusters in the gas phase. Previous DFT studies have predicted that pure Ni n clusters in the intermediate size range ( n =6-10) prefer a pentagonal pyramid growth pattern[78-82]. This infers that the sulfur atoms exert a tremendous influence on the structures of core Ni n clusters when S atoms interact with the Ni n clusters. Since the Ni n cluster has similar structures as the S n cluster, the shell S n cluster also has the atom-by-atom growth principle which is also different from the pure S n cases[83]. The formation pattern studies of Ni n S n clusters elucidate growth processes from NiS cluster to the crystal. Crystalline NiS is dimorphic. The β-NiS millerite structure with trigonal R 3m is found as the stable crystalline phase here, consistent with previous experimental and theoretical results[48-54]. Interestingly, the crystalline β-NiS and its corresponding cluster counterparts have the common core-shell structure as shown in Figure 3. The unit cell of β-NiS displays three S atoms sitting around a Ni 3 triangle. Besides, the self-similarity character and nested-assembly process are also seen in the cluster structures of crystalline NiS. The average bond lengths are also calculated for the binary clusters and shown in Figure S2 along with the shortest Ni-S bond length. It is found that the average bond length increases monotonously with the number of atoms and gradually approaches the bond length of the β-NiS bulk phase. At n =10, the average Ni-S bond length of Ni 10 S 10 cluster is 2.22 Å, very close to 2.28 Å, in the β-NiS crystal. It is worth mentioning that in general, clusters have little or no common structural features with the bulk phases. However, the present work gives strong evidence of the similarity between the small gaseous Ni n S n clusters and NiS bulk. The finding of the characteristics shared by the clusters and their bulk provides important insights towards the size-evolution mechanism of the geometrical structures from atom to cluster to the condensed state. This is very crucial in the discovery of novel nanomaterials in transition-metal sulfides series. 3.3. Physical Prerequisites for Assembly-Disassembly Structures The assembly-disassembly scheme does not apply to all clusters. The physical prerequisites for such a mechanism deserve a thorough investigation to simplify future functional cluster predictions. In this subsection, we figure out the requirements of such a scheme through elucidations of molecular orbital hybridizations in Ni n S n clusters. We start with an exploration of the seeding structure of the Ni 3 S. A building block from the Ni 3 S is noticed in 3D Ni n S n clusters as shown in Figure. 1. Each trigonal pyramid has three Ni-S bonds with the S-capped Ni 3 triangles. Maximizing the coordination number of Ni-S and Ni-Ni bonds will lead to the high stability of the Ni n S n clusters. A molecular orbital (MO) analysis of the bare Ni 3 S cluster was performed. Figure 4 (a) gives the isolated Ni 3 S cluster and its occupied molecular orbital and it is evident from the MO of Ni 3 S that the S atom bonds to each Ni. Three d z orbitals of Ni atoms are overlapping with the orbitals of S atom, yielding three σ bonds featured in MO. The orbital stretching of bare Ni 3 S varies in the Ni n S n systems subjected to additional bonds to deficient or excessive atoms. Table 1 tabulates charge contributions from 3s, 3p, and 3d atomic orbitals on each S atom in Ni n S n clusters with Mulliken Population analysis. A significant amount of charge is retained in the 3d orbital of S atom. The charge occupations in 3d orbital of the Ni n S n clusters at n ≥4 are significantly larger than these of the planar structure at n ≤3. Noteworthy, the planarly-structured Ni 1 S 1 , Ni 2 S 2 , and Ni 3 S 3 clusters cannot form the structure of the Ni 3 S trigonal pyramid, which is the characteristic structural feature of 3D Ni n S n clusters. Combining the charge distribution and structure characteristics, we infer that the S atom has a certain sp 3 d 2 type hybridization involving 3s, 3p, and 3d orbitals in the 3D Ni n S n clusters[84-86]. It is similar to the case in an isolated F 6 S molecule where the d orbitals from sulfur are well known to participate in the hybridization[87,88]. Such an orbital hybridization scheme is in charge of spatial stretching of the clusters that lead to the assembly-disassembly feature eventually. Table 1 Charge distribution in 3s, 3p, and 3d orbitals of per S atom of the Ni n S n system from Mulliken Population analysis. Systems Orbitals 3s 3p 3d Ni 1 S 1 1.890e 4.221e 0.051e Ni 2 S 2 1.834e 4.212e 0.100e Ni 3 S 3 1.835e 4.213e 0.088e Ni 4 S 4 1.811e 4.123e 0.176e Ni 5 S 5 1.812e 4.102e 0.176e Ni 6 S 6 1.812e 4.101e 0.171e Ni 7 S 7 1.812e 4.096e 0.177e Ni 8 S 8 1.814e 4.022e 0.178e Ni 9 S 9 1.805e 4.034e 0.177e Ni 10 S 10 1.802e 4.001e 0.179e Ni 3 S 1 1.773e 4.104e 0.190e Ni 6 S 1 1.697e 3.446e 0.524e To verify this hypothesis, we optimized the structure of the F 6 S cluster and compared it with other Ni 6 X (X refers to the chalcogenide element of O, S, Se) and the Zn 6 S clusters. Different elements of chalcogen were selected to have an expanded view of the sp 3 d 2 type hybridization in these similar clusters. The Zn atom is considered to analyses the effects of the interactions and bonding modes with various d orbitals in the TM participants compared with Ni. The geometric structures and corresponding MOs are depicted in Figure. 4(b). The octahedral F 6 S has six fluorine atoms surrounding the central sulfur atom, while the Ni 6 S is a trigonal prismatic cluster with six nickel atoms surrounding the central sulfur atom. This is rational because there is an insufficient sp 3 d 2 orbital hybridization of the S atom in the Ni 6 S cluster. As a result, the octahedral geometry is distorted and displays low symmetry. In F 6 S, six sp 3 d 2 hybrid bond orbitals on the sulfur are overlapped by six fluorine 2p orbitals, yielding six σ (sigma) bonds. The MOs of Ni 6 S show that there are six equal Ni-S bonds with equal strength and length. The sulfur atom forms covalent S-Ni bonds with six surrounding nickels by d z -sp 3 d 2 overlap as shown in Figure 4(b). This indicates overlapping between Ni atom and S effectively yields Ni 6 S cluster with trigonal prismatic structure. The overall stability of the system is enhanced. These results are consistent with our previous discussions of the structures and stabilities of Ni n S n clusters. According to our structural postulate and analysis, Ni 6 Se will also exhibit the same structure and MO graph since Se and S have similar electronic configurations, in agreement with our MO calculations. However, such σ d z -sp 3 d 2 overlap scheme is not suitable for the Zn 6 S case because d orbital is fully occupied by the valence electrons in IB and IIB groups and nearly no empty d orbitals can be provided for forming covalent Zn-S bonds. As clearly seen in the Zn 6 S, s orbitals in Zn participate in σ s–sp 3 d 2 overlap, which is different from the Ni 6 X cases where the d z orbitals of Ni bond to the X elements. On the other hand, the trigonal prismatic structure of Ni 6 O failed to reach structural optimization due to the large energy difference between the relevant 2p (filled) and 3d (empty) orbitals in the oxygen atom. Consequentially, the outermost 2p electron can hardly jump to the 3d orbital to achieve sp 3 d 2 hybridization, which leads to the absent trigonal prismatic structure in the Ni 6 O cluster. From the results, it was concluded that the disassembly and assembly core-shell structural motif might be attributed to the certain sp 3 d 2 type hybridization in the trigonal pyramid structure Ni 3 S, which is the principal structural unit in the formation of Doll-like structures of Ni n S n clusters. Such a hybrid mechanism may be useful in identifying new nanomaterials with binary self-similar structures. 3.4. Stability Assessment The stability of clusters is evaluated by the average binding energy ( E b ) per atom and the second-order energy difference (Δ 2 E ) of the Ni n S n clusters and β-NiS bulk phase. The corresponding formulas are given below: where E (Ni n S n ), E (Ni) and E (S) represent the total energy of the NinSn cluster, and the energies of a single Ni and S atom, respectively. In Figure 5, the E b of Ni n S n cluster for the ground state structures increases monotonically with the size, from 2.27 eV in Ni 1 S 1 to 4.05 eV in Ni 1 0 S 1 0 . This indicates an energetically favorable process in the cluster growth. In comparison, we also calculated the cohesive energy of NiS bulk phase. The calculated cohesive energies are 5.26 eV and 5.24 eV respectively for β-NiS and α-NiS in a unit cell under low-temperature conditions, slightly larger than other DFT values [52]. Although the binding energy per atom for n =10 cluster is significantly smaller than the cohesive energy of bulk β-NiS, one can speculate that the E b of Ni n S n cluster will gradually approach the value of solid as the cluster grows toward the bulk. It is noteworthy that the E b of the binary transition metal sulfide Ni n S n clusters is significantly higher than that of pure S 2 n and Ni 2 n clusters at the same cluster sizes[81-83]. For example, the calculated E b for Ni 2 S 2 is 3.28 eV, which is 1.41 eV and 0.98 eV higher than those of Ni 4 (1.87 eV)[82] and S 4 clusters (2.30 eV)[83]. Ni-S interactions are energetically favorable leading to the high stabilization of the binary clusters, suggesting the Ni with partially filled d shells have strong interactions with S atom with unsaturated p electrons. In addition to the average binding energy, we have calculated the second-order energy difference (Δ₂ E ) to further evaluate the relative stability of Ni n S n clusters. Figure 5 shows the variation of the Δ 2 E with respect to n for the clusters. From the figure, it is evident that for the Ni n S n clusters, the Δ 2 E exhibits a clear odd-even oscillation pattern, with peaks occurring at n =2, 5, 7, and 9. This suggests that the corresponding clusters have relatively high stability (compared to their neighbors), which is consistent with the conclusions drawn from the binding energy analysis earlier. The thermodynamic stabilities of the Ni n S n clusters for the ground-state structure were also examined using molecular dynamics simulation (MD) at fixed volume with a thermostat to maintain a constant temperature ensemble (NVT) implemented in the CASTEP code at room temperature (T=300 K). The Nosé-Hoover-Langevin (NHL) thermostat was used for NVT ensemble[89]. Figure 6 plots the MD results for the Ni 5 S 5 and Ni 6 S 6 clusters. Within 20 ps , changes were undetected in any structures while the instantaneous values of the relative potential energy fluctuate due to thermal fluctuations, but the average values stay constant. Thus, the ground-state clusters of these Ni n S n clusters are real and stable at room temperature. 3.5. Electronic and Magnetic Properties The electronic structures of Ni n S n clusters are studied. The deformation electron density (DED) of the Ni 6 S 6 cluster is plotted in Figure 7(a) as an example. Electron density is accumulated around midway of the Ni-S contacts, showing covalent characteristics and robust Ni-S interactions. The interaction between the S-S atoms is negligible. Figure 7(b) depicts the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) pictures of the starting NiS dimer to show electronic structures. In the figure, the electron density of the HOMO and the LUMO states are not only mainly distributed around each Ni and S atoms, but also obviously in the middle of Ni and S atoms. Molecular orbital analysis indicates strong hybridization between the d orbitals of Ni and the p orbitals of S. This phenomenon is also reflected in the electronic partial density of states (PDOS) shown in Figure 7(c). The calculated PDOS listed in Figure 7(c) shows that there is an obvious superposition of sharp peaks originating from the d orbital of Ni and p orbital of S from -6 eV to 6 eV energy region near the Fermi level. The average atomic charge (e) on S atom and HUMO-LUMO gap or bandgap are studied for Ni n S n lowest-energy clusters ( n =1-10) and β-NiS bulk phase. Figure 8a shows that charge transfer occurs from Ni to S for each cluster due to a larger electronegativity of the S element. As a result, polarity develops in Ni-S interactions. The mixed ionic-like and covalent bonding are responsible for the high stability of these Ni n S n clusters. In the β-NiS crystalline phase, the charge transfer was 0.093 e/atom, which reveals the charge transfer of clusters is trending toward the bulk parameter with increasing cluster size. A similar trend is also found in the energy gap as shown in Figure 8b. The overall gap energy decreases with n , reaches 0.236 eV at n =10, and eventually vanishes in the crystalline NiS. It is worth noting that there is a peak at n =4 for the gap in Figure 8b. This is due to the structural transition from the 2D to 3D of the Ni n S n clusters. Consequently, the larger clusters approximate closer physicochemical properties concerning the bulk. Table 2 The calculated vIP, vEA and chemical hardness (ƞ) of the Ni n S n system. Systems vIP (eV) vEA (eV) ƞ Ni 1 S 1 8.79 0.61 4.09 Ni 2 S 2 8.14 1.46 3.34 Ni 3 S 3 8.37 2.93 2.72 Ni 4 S 4 7.19 1.54 2.82 Ni 5 S 5 7.16 1.89 2.64 Ni 6 S 6 7.22 2.36 2.43 Ni 7 S 7 7.31 2.85 2.23 Ni 8 S 8 7.25 3.29 1.98 Ni 9 S 9 7.17 3.24 1.97 Ni 10 S 10 6.90 3.41 1.75 To further analyze the electronic stability of the clusters, we have calculated the vertical ionization potential (vIP) and vertical electron affinity (vEA) parameters. The equations and definitions can be found in references [90,91]. The variations of the vIP and vEA parameters as functions of cluster size are shown in Table 2. It is observed that the vIP generally decreases as the cluster size increases, while the vEA first increases for cluster sizes n = 1–3, reaching a peak of 2.93 at Ni 3 S 3 . From Ni 4 S 4 onward, the vEA again increases gradually. To analyze the chemical stability of the clusters, we determined the chemical hardness (ƞ), which can be derived from the vIP and vEA values. It is found that ƞ decreases monotonically with increasing cluster size, suggesting that larger clusters evolve toward a metallic-like electronic structure. These conclusions are consistent with the findings of Rodríguez-Kessler et al. in their study of Pt n Cu n binary clusters [92]. Magnetic properties of the predicted clusters are different from the naked Ni peers. Although previous experimental and DFT studies showed that the naked Ni n clusters (without stabilizing ligands ) hold the robust magnetic properties[81,93,94], our work predicted coating with shell S n cluster will reduce or even eliminate the magnetic properties/magnetism of the core Ni n clusters. This is indeed in line with the magnetic properties of the Ni-S systems at a squeezed space[56]. Table 3 shows that the magnetic moment of Ni atoms is completely quenched in the Ni n S n system, except for the dimer NiS and planar triangle Ni 3 S 3 clusters holding magnetic moments of 2 µ B and 4 µ B respectively. The general trend converges to the non-magnetic NiS bulk phase. Similar behavior has been observed in the case of Ni sitting in the center of an Au 6 ring, wherein, the magnetic property of Ni impurity-doped Au 6 cluster is completely quenched[95]. The Ni n S n clusters may not be good candidates for magnetics despite their structural uniqueness. Table 3 Total and local magnetic moment (in µ B ) for the Ni n S n and Ni n core clusters calculated in this work with the comparison of the bare Ni n clusters from the experiment and other DFT results. The hyphen (-) means no reference data. a) Ref.[97], b) Ref.[98] This work Experiment a) Other DFT b) Cluster size ( n ) Ni n S n Core Ni n Bare Ni n n =1 2 1.100 2 2 n =2 0 0 - 3 n =3 4 2.996 - 4 n =4 0 0 - 5 n =5 0 0 9 4 n =6 0 0 9 8 n =7 0 0 11 8 n =8 0 0 13 8 n =9 0 0 12 8 n =10 0 0 13 8 3.6. Potential Applications in Hydrogen Storage The sulfur atom in the trigonal pyramidal Ni₃S motif exhibits a tendency toward sp³d² hybridization in the predicted 3D Ni n S n clusters. This hybridization leaves the S atom with three essentially non-bonding orbitals, manifesting as dangling bonds in the isolated Ni₃S unit. These sites serve as potential active centers for hydrogen anchoring. Figure 4(c) shows the optimized structure of a Ni₃S cluster adsorbing three hydrogen atoms, along with its corresponding molecular orbital (MO) diagram. The results indicate that each hydrogen atom donates one electron to form a stable bond with the central S atom, which in turn bonds covalently with three Ni atoms and three H atoms simultaneously. The hydrogen storage capacity of Ni n S n clusters is significantly enhanced compared to their individual Ni₃S building blocks, owing to the cumulative effect of multiple such units. To explore this phenomenon further, we selected the Ni₄S₄ cluster as a representative system for systematic hydrogen adsorption analysis. Figure 9 presents the optimized geometries of Ni₄S₄ clusters adsorbing from 1 to 16 hydrogen atoms. Detailed Cartesian coordinates of these configurations are provided in Table S3 in the Supporting Information. During the simulations, hydrogen atoms were incrementally introduced and allowed to fully relax along with the cluster structure until saturation was reached. Our results reveal that chemisorbed hydrogen atoms can bind not only to Ni atoms but also to S atoms. However, Ni atoms exhibit a significantly stronger affinity for hydrogen adsorption than S atoms, as evidenced by the comparison of adsorption energies. For instance, when a single hydrogen atom is adsorbed, the configuration with H bound to a Ni atom is notably more stable than that with H bound to a S atom. Furthermore, when hydrogen atoms are adsorbed on Ni sites, the Ni₄S₄ cluster retains its characteristic square antiprism geometry with only minimal structural deformation. In contrast, adsorption on S sites leads to more substantial distortions. As more hydrogen atoms are adsorbed, the Ni–S bond lengths increase progressively, and the square antiprism framework becomes increasingly unstable. Beyond 12 hydrogen atoms, the structure approaches the point of collapse, indicating a sharp decline in structural integrity. According to our calculations, Ni₄S₄ can stably adsorb up to 16 hydrogen atoms. Attempts to adsorb more than this result in structural convergence failure or complete disintegration into a disordered configuration. The gravimetric hydrogen storage capacity of the Ni₄S₄ cluster was calculated using the following formula: where n is the number of hydrogen atoms, M H is the atomic mass of hydrogen, and M cluster is the total mass of the pristine cluster. As summarized in Table 4, the hydrogen storage capacity increases from 0.3 wt% for a single H atom to a maximum of 4.3 wt% for 16 H atoms. Although this value does not yet meet the U.S. Department of Energy (DOE) 2025 target of 5.5 wt% for onboard hydrogen storage systems, it shows promising potential. Notably, recent work by Rodríguez-Kessler et al. has highlighted the viability of binary clusters as efficient hydrogen storage media, which supports the conclusions drawn in our study regarding the application potential of Ni n S n clusters in clean energy technologies [96–99]. To further understand the nature of hydrogen-cluster interactions, we analyzed the average bond lengths between hydrogen and Ni/S atoms, along with Mulliken charge distributions on the adsorbed hydrogen atoms. As shown in Table 4, the average H–S bond length increases from 1.39 Å to 1.47 Å, while the H–Ni bond length grows from approximately 1.47 Å to 1.60 Å as the number of hydrogen atoms increases. This bond elongation suggests weakened chemisorption strength at higher hydrogen coverage. Mulliken charge analysis also shows a decrease in charge transfer from the cluster to hydrogen atoms with increased loading, corroborating the observed trend in reduced interaction strength. Table 4 Calculated hydrogen weight percentage (wt%), average bond distances (Å) of H–Ni and H–S, and average Mulliken charge (a.u.) on the hydrogen atom for the Ni₄S₄ cluster adsorbing 1 to 16 hydrogen atoms. A hyphen (–) indicates unavailable data. Number of the H atoms Hydrogen weight percent Bond length Charge H-Ni H-S H-Ni H-S 1 0.3% 1.471 - 0.088 - 2 0.6% 1.472 - 0.085 - 3 0.8% 1.472 1.393 0.080 0.233 4 1.1% 1.473 1.394 0.075 0.221 5 1.4% 1.476 1.396 0.073 0.218 6 1.7% 1.478 1.397 0.069 0.208 7 1.9% 1.479 1.398 0.070 0.210 8 2.2% 1.483 1.401 0.062 0.206 9 2.5% 1.486 1.409 0.058 0.201 10 2.8% 1.492 1.412 0.059 0.186 11 3.0% 1.510 1.425 0.055 0.171 12 3.3% 1.530 1.432 0.048 0.162 13 3.6% 1.552 1.447 0.045 0.153 14 3.8% 1.575 1.452 0.042 0.146 15 4.1% 1.583 1.469 0.037 0.138 16 4.3% 1.596 1.473 0.035 0.129 4. CONCLUSIONS AND SUMMARY In conclusion, a successful disassembly-assembly strategy has been revealed in the structural optimizations of the binary Ni n S n ( n =1-10) clusters and paved the way to study cluster properties. Based on the DFT results, the assembling paths of Ni n S n systems are simplified in inlaying the core Ni n to shell S n clusters within the same geometric point group. Following this discovery, investigations of the structural evolution of the compound clusters are reduced to probations of component Ni n or S n systems with simpler structures. The NiS cluster can follow the core-shell constructions to reach the crystalline phase. The specific orbital hybridization from the bonding S atoms is proposed in charge of the spatial extension of the clusters, and the assembly-disassembly scheme for the TMD clusters. Despite magnetic mediocrity, the thermostable Ni n S n clusters are endowed with strong abilities to absorb hydrogen atoms for clean energy storage. Starting from the explicated NiS case, the present work is hoped to benefit the overall cluster sciences by serving a fast and efficient shortcut to identifying the most probable isomers of large clusters complicated in structure and rich in element species. Declarations Author Contribution Weide Liu, Fenfa Zhang, Bowei Jin, Xuezhou Lu, and Zhengyang Ding wrote the main manuscript text. Xiao Wang and Meng Zhang contributed to the conceptualization and manuscript revision. All authors reviewed and approved the final manuscript. Supporting Information Available: The benchmark calculation on NiS dimer using a different type of functionals. Details of structures and x,y,z coordinates of the lowest-energy structures and low-lying isomers of Ni n S n ( n =1-10) clusters, as well as the Ni 4 S 4 cluster with adsorbed hydrogen atoms. The shortest and average Ni-S bond lengths of the Ni n S n clusters ( n =1-10) and β-NiS bulk phase. These materials are available free of charge via the Internet. Notes The authors declare no competing financial interest. ACKNOWLEDGMENTS This work is financially supported by the University Student Innovation Program of China (Grant No. 201810251078). References Rohmer M. M.; Bénard M.; Poblet J. M. Structure, reactivity, and growth pathways of metallocarbohedrenes M8C12 and transition metal/carbon clusters and nanocrystals: A challenge to computational chemistry. Chem. Rev. 2000 , 100 , 495-542. Einax M.; Dieterich W.; Maass P. Colloquium: Cluster growth on surfaces: Densities, size distributions, and morphologies. Rev. Mod. Phys. 2013 , 85 , 921. Pauna H.; Shi X.; Huttula M.; et al. Evolution of lithium clusters to superatomic Li 3 O+. Appl. Phys. Lett. 2017 , 111 , 103901. Zubarev D. Y.; Averkiev B. B.; Zhai H. J. et al. Aromaticity and antiaromaticity in transition-metal systems. Phys. Chem. Chem. Phys. 2008 , 10 , 257-267. Kroto H. W.; Heath J. R.; O'Brien S. C.; et al. C60: Buckminsterfullerene. Nature 1985 , 318 , 162-163. Reveles J. U.; Clayborne P. A.; Reber A. C.; et al. Designer magnetic superatoms. Nat. Chem., 2009 , 1 , 310. Wang X.; Adeleke A. A.; Cao W.; et al. Structures of Nanoalloy Clusters Au n Al n (n=1–10) and the Growth Patterns to the Bulk Phase. J. Phys. Chem. C, 2016 , 120 , 25588-25595. Gong X. G. Structure and stability of cluster-assembled solid Al12C(Si): a first-principles study. Phys. Rev. B, 1997 , 56 , 1091. Wang L. M.; Wang L. S. Probing the electronic properties and structural evolution of anionic gold clusters in the gas phase. Nanoscale 2012 , 4 , 4038-4053. Heine T. Transition metal chalcogenides: ultrathin inorganic materials with tunable electronic properties. Accounts. Chem. Res. 2015 , 48 , 65-72. Soldatov A. V.; Kravtsova A. N.; Fleet M. E.; et al. Electronic structure of MeS (Me= Ni, Co, Fe): x-ray absorption analysis. J. Phys-Condens. Mat. 2004 , 16 , 7545. Chianelli R. R.; Daage M.; Ledoux M. J. Fundamental studies of transition-metal sulfide catalytic materials, Advances in Catalysis. Academic Press, 1994 , 40 , 177-232. Rohrbach A.; Hafner J.; Kresse G. Electronic correlation effects in transition-metal sulfides. J. Phys-Condens. Mat. 2003 , 15 , 979. Chianelli R. R.; Berhault G.; Torres B. Unsupported transition metal sulfide catalysts: 100 years of science and application. Catal. Today. 2009 , 147 , 275-286. El Nakat J. H.; Dance I. G.; Fisher K. J.; et al. Laser-ablation FTICR (Fourier-transform ICR) mass spectrometry of metal sulfides: gaseous anionic nickel-sulfur [Ni x S y ] clusters. J. Am. Chem. Soc. 1991 , 113 , 5141-5148. El Nakat J. H.; Dance I. G.; Fisher K. J.; et al. Gas phase copper chalcogenide cluster ions, formed by laser-ablation. Inorg. Chem. 1991 , 30 , 2957-2958. El Nakat J.; Dance I.; Fisher K.; et al. Gas-phase silver chalcogenide ions investigated by laser-ablation Fourier transform ion cyclotron resonance mass spectrometry. J. Chem. Soc., Chem. Commun. 1991 , 11 , 746-748. El Nakat J. H.; Fisher K. J.; Dance I. G.; et al. Gas phase metal chalcogenide cluster ions: a new cobalt-sulfur [CoxSy]-series up to [Co 38 S 24 ]-and two iron-sulfur [Fe x S y ]-series. Inorg. Chem. 1993 , 32 , 1931-1940. Dance I. G.; Fisher K. J.; Willett G. D. Reactions of 29 transition metal cations, in the same oxidation state and under the same gas-phase conditions, with sulfur. Inorg. Chem. 1996 , 35 , 4177-4184. Fisher K.; Dance I.; Willett G.; et al. Gas-phase metal sulfide cluster anions. J. Chem. Soc., Dalton Trans. 1996 , 5 , 709-718. Dance I. G,; Fisher K. J.; Willett G. D. Molecular manganese sulfide clusters formed by laser ablation. J. Chem. Soc., Dalton Trans. 1997 , 15 , 2557-2562. Zhang N.; Yu Z.; Wu X.; et al. Formation and photolysis of tantalum sulfide cluster ions. J. Chem. Soc., Faraday Trans. 1993 , 89 , 1779-1782. Yu Z.; Zhang N.; Wu X.; et al. The production and photodissociation of iron–sulfur cluster ions. J. Chem. Phys. 1993 , 99 , 1765-1770. Shi Y.; Zhang N.; Gao Z.; et al. The formation, photodissociation, and bond structure of cobalt–sulfur cluster ions. J. Chem. Phys. 1994 , 101 , 9528-9533. Liu Y.; Xing J.; Fu H.; et al. Structural stability, mechanical properties, electronic structures and thermal properties of XS (X= Ti, V, Cr, Mn, Fe, Co, Ni) binary compounds. Phys. Lett. A 2017 , 381 , 2648-2657. Liang B.; Andrews L. Infrared spectra and density functional theory calculations of group 4 transition metal sulfides. J. Phys. Chem. A 2002 , 106 , 6295-6301. Liang B.; Andrews L. Matrix Infrared Spectra and Quasirelativistic DFT Studies of ThS and ThS 2 . J. Phys. Chem. A 2002 , 106 , 4038-4041. Liang B.; Andrews L. Infrared spectra and density functional theory calculations of group 6 transition metal sulfides in solid argon. J. Phys. Chem. A 2002 , 106 , 6945-6951. Liang B.; Wang X.; Andrews L. Infrared spectra and density functional theory calculations of group 8 transition metal sulfide molecules. J. Phys. Chem. A 2009 , 113 , 5375-5384. Liang B.; Wang X.; Andrews L. Infrared spectra and density functional theory calculations of group 10 transition metal sulfide molecules and complexes. J. Phys. Chem. A 2009 , 113 , 3336-3343. Zhao Y. C.; Yuan J.; Zhang Z, G.; et al. Structures of manganese polysulfides: mass-selected photodissociation and density functional calculations. Dalton Trans. 2011 , 40 , 2502-2508. Kenzler S.; Schrenk C.; Frojd A. R.; et al. Au 70 S 20 (PPh 3 ) 12 : an intermediate sized metalloid gold cluster stabilized by the Au 4 S 4 ring motif and Au-PPh 3 groups. Chem. Commun. 2018 , 54 , 248-251. Pei Y.; Gao Y.; Shao N.; et al. Thiolate-protected Au 20 (SR) 16 cluster: prolate Au 8 core with new [Au 3 (SR) 4 ] staple motif . J. Am. Chem. Soc. 2009 , 131 , 13619-13621. Pei Y.; Tang J.; Tang X.; et al. New structure model of Au 22 (SR) 18 : bitetrahederon golden kernel enclosed by [Au6(SR)6]Au(I) complex. J. Phys. Chem. Lett. 2015 , 6 , 1390-1395. Mondal A.; Vomiero A. 2D transition metal dichalcogenides-based electrocatalysts for hydrogen evolution reaction. Adv. Funct. Mater. 2022 , 32 (52), 2208994. Zhang J.; Wu J.; Guo H.; et al. Unveiling active sites for the hydrogen evolution reaction on monolayer MoS₂. Adv. Mater. 2017 , 29 (42), 1701955. Voiry D.; Salehi M.; Silva R.; et al. Conducting MoS₂ nanosheets as catalysts for hydrogen evolution reaction. Nano Lett. 2013 , 13 (12), 6222-6227. Li G.; Zhang D.; Qiao Q.; et al. All the catalytic active sites of MoS₂ for hydrogen evolution. J. Am. Chem. Soc. 2016 , 138 (51), 16632-16638. Cao Y. Roadmap and direction toward high-performance MoS₂ hydrogen evolution catalysts. ACS Nano 2021 , 15 (7), 11014-11039. Li Y.; Wang H.; Xie L.; et al. MoS₂ nanoparticles grown on graphene: an advanced catalyst for the hydrogen evolution reaction. J. Am. Chem. Soc. 2011 , 133 (19), 7296-7299. Cheng Y.; Lu S.; Liao F.; et al. Rh-MoS₂ nanocomposite catalysts with Pt-like activity for hydrogen evolution reaction. Adv. Funct. Mater. 2017 , 27 (23), 1700359. Yu E.; Pan Y. Enhancing the catalytic hydrogen evolution reaction (HER) of the defective borophene@Pt/Pd/MoS₂ heterojunction. Int. J. Hydrogen Energy 2024 , 50 , 920-931. Duan H.; Wang C.; Li G.; et al. Single-atom-layer catalysis in a MoS₂ monolayer activated by long-range ferromagnetism for the hydrogen evolution reaction: beyond single-atom catalysis. Angew. Chem. 2021 , 133 (13), 7327-7334. Gibbs G.; V.; Downs R. T.; Prewitt C. T.; et al. Electron Density Distributions Calculated for the Nickel Sulfides Millerite, Vaesite, and Heazlewoodite and Nickel Metal: A Case for the Importance of Ni− Ni Bond Paths for Electron Transport. J. Phys. Chem. B 2005 , 109 , 21788-21795. Metcalf P. A.; Crooker B. C.; McElfresh M.; et al. Low-temperature electronic and magnetic properties of single-crystal Ni 3 S 2 . Phys. Rev. B 1994 , 50 , 2055. Tyler J. M.; Fry J. L. Energy bands of hexagonal NiS. Phys. Rev. B 1970 , 1 , 4604. Kullerud G.; Yund R. A. The Ni-S system and related minerals. J. Petrol. 1962 , 3 , 126-175. Sowa H.; Ahsbahs H.; Schmitz W. X-ray diffraction studies of millerite NiS under non-ambient conditions. Phys. Chem. Miner. 2004 , 31 , 321-327. Goh S. W.; Buckley A. N.; Lamb R. N.; et al. Sulfur electronic environments in α-NiS and β-NiS: examination of the relationship between coordination number and core electron binding energies. Phys. Chem. Miner. 2006 , 33 , 98-105. Krishnakumar S. R.; Shanthi N.; Sarma D. D. Electronic structure of millerite NiS. Phys. Rev. B 2002 , 66 , 115105. Wang H.; Pring A.; Ngothai Y.; et al. The kinetics of the α→ β transition in synthetic nickel monosulfide. Am. Mineral. 2006 , 91 , 171-181. Raybaud P.; Kresse G.; Hafner J.; et al. Ab initio density functional studies of transition-metal sulphides: I. Crystal structure and cohesive properties. J. Phys-Condens. Mat. 1997 , 9 , 11085. Raybaud P.; Hafner J.; Kresse G.; et al. Ab initio density functional studies of transition-metal sulphides: II. Electronic structure. J. Phys-Condens. Mat. 1997 , 9 , 11107. Wang J. H.; Cheng Z.; Brédas J. L.; et al. Electronic and vibrational properties of nickel sulfides from first principles. J. Chem. Phys. 2007 , 127 , 214705. Musselman I. H.; Linton R. W.; Simons D. S. Cluster ion formation under laser bombardment. Studies of recombination using isotope labeling. Anal. Chem. 1988 , 60 , 110-114. Shi X.; Posysaev S.; Huttula M.; et al. Metallic contact between MoS 2 and Ni via Au nanoglue. Small 2018 , 14 , 1704526. Deng S.; Zhang K.; Xie D.; et al. High-index-faceted Ni3S2 branch arrays as bifunctional electrocatalysts for efficient water splitting. Nano-Micro. Lett. 2019 , 11 , 12. Panda C.; Menezes P. W.; Yao S.; et al. Boosting Electrocatalytic Hydrogen Evolution Activity with a NiPt 3 @NiS Heteronanostructure Evolved from a Molecular Nickel–Platinum Precursor. J. Am. Chem. Soc. 2019 , 141 , 13306-13310. Wang Y.; Lv J.; Zhu L.; et al. CALYPSO: A method for crystal structure prediction. Comput. Phys. Commun. 2012 , 183 , 2063-2070. Wang Y.; Miao M.; Lv J.; et al. An effective structure prediction method for layered materials based on 2D particle swarm optimization algorithm. J. Chem. Phys. 2012 , 137 , 224108. Lv J.; Wang Y.; Zhu L.; et al. Particle-swarm structure prediction on clusters. J. Chem. Phys. 2012 , 137 , 084104. Frisch M. J.; Trucks G. W.; Schlegel H. B.; et al. Gaussian 09. vol. Revision D. 01, Gaussian. Inc., Wallingford, CT 2009 . Stevens W. J.; Basch H.; Krauss M. Compact effective potentials and efficient shared‐exponent basis sets for the first‐and second‐row atoms. J. Chem. Phys. 1984 , 81 , 6026-6033. Hay P. J.; Wadt W. R. Ab initio effective core potentials for molecular calculations. Potentials for the transition metal atoms Sc to Hg. J. Chem. Phys. 1985 , 82 , 270-283. Delley B. An all‐electron numerical method for solving the local density functional for polyatomic molecules. J. Chem. Phys. 1990 , 92 , 508-517. Perdew J. P.; Burke K.; Ernzerhof M. Generalized gradient approximation made simple. Phys. Rev. Lett. 1996 , 77 , 3865. Becke A. D. Density-functional exchange-energy approximation with correct asymptotic behavior. Phys. Rev. Appl. 1988 , 38 , 3098. Lee C.; Yang W.; Parr R. G. Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density. Phys. Rev. B 1988 , 37 , 785. Perdew J. P.; Wang Y. Accurate and simple analytic representation of the electron-gas correlation energy. Phys. Rev. B 1992 , 45 , 13244. Huber K. P. Molecular spectra and molecular structure: IV. Constants of diatomic molecules. Springer Science & Business Media 2013 . Morse M. D.; Hansen G. P.; Langridge‐Smith P. R. R.; et al. Spectroscopic studies of the jet‐cooled nickel dimer. J. Chem. Phys. 1984 , 80 , 5400-5405. Kant A. Dissociation energies of diatomic molecules of the transition elements. I. Nickel. J. Chem. Phys. 1964 , 41 , 1872-1876. Zhang M.; Zhang J.; Gu T.; et al. First-principles investigations of chirality in trimetallic alloy clusters: AlMnAu n (n=1–7). J. Phys. Chem. A 2015 , 119 , 3458-3470. Elgavi H.; Krekeler C.; Berger R.; et al. Chirality in copper nanoalloy clusters. J. Phys. Chem. C 2011 , 116 , 330-335. Dance I. G.; Fisher K. J. Density functional calculations of electronic structure, geometric structure and stability for molecular manganese sulfide clusters. J. Chem. Soc., Dalton Trans. 1997 , 15 , 2563-2576. Ming F.; Wang K.; Pan S.; et al. Assembling and disassembling Ag clusters on Si (111)-(7×7) by vertical atomic manipulation. ACS nano 2011 , 5 , 7608-7616. Wu X.; Liang X.; Du Q.; et al. Medium-sized (n=14–20) clusters: a combined study of photoelectron spectroscopy and DFT calculations. J. Phys-Condens Mat. 2018 , 30 , 354002. Cleveland C. L.; Landman U. The energetics and structure of nickel clusters: Size dependence. J . Chem. Phys . 1991 , 94 , 7376-7396. Alonso J. A. Electronic and atomic structure, and magnetism of transition-metal clusters. Chem. Rev. 2000 , 100 , 637-678. Grigoryan V. G. Springborg M. Structural and energetic properties of nickel clusters: 2⩽N⩽ 150. Phys. Rev. B 2004 , 70 , 205415. Deshpande M.; Kanhere D. G.; Pandey R. Structures, energetics, and magnetic properties of Ni n B clusters with n= 1–8, 12. Phys. Rev. A 2005 , 71 , 063202. Menon M.; Connolly J.; Lathiotakis N.; et al. Tight-binding molecular-dynamics study of transition-metal clusters. Phys. Rev. B 1994 , 50 , 8903. Jin Y.; Maroulis G.; Kuang X.; et al. Geometries, stabilities and fragmental channels of neutral and charged sulfur clusters: S n Q (n= 3–20, Q= 0,±1). Phys. Chem. Chem. Phys. 2015 , 17 , 13590-13597. Streitwieser A. Molecular orbital theory for organic chemists. New York: Wiley, 1961 . Jung J.; Kim H.; Han Y. K. Can an electron-shell closing model explain the structure and stability of ligand-stabilized metal clusters?. J. Am. Chem. Soc. 2011 , 133 , 6090-6095. Petrucci R. H.; Harwood W. S.; Herring F. G. General chemistry: principles and modern applications. Prentice Hall, 2002 . Gillespie R. J. The VSEPR model revisited. Chem. Soc. Rev. 1992 , 21, 59-69. Gillespie R. J. The valence-shell electron-pair repulsion (VSEPR) theory of directed valency. J. Chem. Educ. 1963 , 40, 295. Leimkuhler B.; Noorizadeh E.; Theil F. A gentle stochastic thermostat for molecular dynamics. J. Stat. Phys. 2009 , 135 , 261-277. Rodríguez-Carrera S.; Rodríguez-Kessler P L.; Ambriz-Vargas F. et al. First principles study for Ag-based core-shell nanoclusters with 3d-5d transition metal cores for the oxygen reduction reaction. Inorg. Chim. Acta 2024 , 572 , 122301. Rodríguez-Kessler P. L.; Muñoz-Castro A.; Alonso-Dávila P. A. et al. Structural, electronic and catalytic properties of bimetallic Pt n Ag n ( n = 1–7) clusters. J. Alloys Compd. 2020 , 845 , 155897. Rodríguez-Kessler P. L.; Rodríguez-Domínguez A. R.; Muñoz-Castro A. On the structure and reactivity of Pt n Cu n ( n = 1–7) alloy clusters. Phys. Chem. Chem. Phys. 2021 , 23 (12), 7233-7239. Apsel S. E.; Emmert J. W.; Deng J. et al. Surface-enhanced magnetism in nickel clusters. Phys. Rev. Lett. 1996 , 76 , 1441. Song W.; Lu W. C.; Wang C. Z.; et al. Magnetic and electronic properties of the nickel clusters Ni n (n⩽30). Comput . Theor . Chem . 2011 , 978 , 41-46. Zhang M.; He L. M.; Zhao L. X.; et al. Tuning magnetic moments by 3d transition-metal-doped Au6 Clusters. J . Phys . Chem . C 2009 , 113 , 6491-6496. Rodríguez-Kessler P. L.; Rodríguez-Carrera S.; Muñoz-Castro A. Hydrogen evolution reaction on Ag 12 M icosahedrons: A DFT study. Chem. Phys. Lett. 2024 , 856 , 141588. Olalde-López D.; Rodríguez-Kessler P. L.; Rodríguez-Carrera S.; et al. Hydrogen storage properties for bimetallic doped boron clusters M 2 B 7 (M= Fe, Co, Ni). Int. J. Hydrogen Energy . 2024. Jana G.; Pan S.; Rodríguez-Kessler P. L.; et al. Adsorption of molecular hydrogen on lithium–phosphorus double-helices. J. Phys. Chem. C 2018 , 122 (49), 27941-27946. Rodríguez-Kessler P. L.; Murillo F.; Rodríguez-Domínguez A. R.; et al. Structure of V-doped Pdn (n= 2–12) clusters and their ability for H 2 dissociation. Int. J. Hydrogen Energy . 2018 , 43 (45), 20636-20644. Additional Declarations No competing interests reported. Supplementary Files Supportinginformation.pdf Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-5906730","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":449283417,"identity":"46611696-5ae4-49bd-a520-5aba93964bec","order_by":0,"name":"Weide Liu","email":"","orcid":"","institution":"East China University of Science and Technology","correspondingAuthor":false,"prefix":"","firstName":"Weide","middleName":"","lastName":"Liu","suffix":""},{"id":449283418,"identity":"5ce0ca87-e7b5-42b0-8fdc-021cada9640e","order_by":1,"name":"Fenfa Zhang","email":"","orcid":"","institution":"East China University of Science and Technology","correspondingAuthor":false,"prefix":"","firstName":"Fenfa","middleName":"","lastName":"Zhang","suffix":""},{"id":449283419,"identity":"33aa1a28-9096-4acb-869a-954196756c97","order_by":2,"name":"Bowei Jin","email":"","orcid":"","institution":"East China University of Science and Technology","correspondingAuthor":false,"prefix":"","firstName":"Bowei","middleName":"","lastName":"Jin","suffix":""},{"id":449283420,"identity":"9d83bd6e-aa8a-4e55-a350-a2dda89ba068","order_by":3,"name":"Xuezhou Lu","email":"","orcid":"","institution":"East China University of Science and Technology","correspondingAuthor":false,"prefix":"","firstName":"Xuezhou","middleName":"","lastName":"Lu","suffix":""},{"id":449283421,"identity":"81ffb675-536e-47db-af51-18776968941e","order_by":4,"name":"Zhengyang Ding","email":"","orcid":"","institution":"East China University of Science and Technology","correspondingAuthor":false,"prefix":"","firstName":"Zhengyang","middleName":"","lastName":"Ding","suffix":""},{"id":449283422,"identity":"d10eccf0-7c85-4451-b581-edb8e41905d6","order_by":5,"name":"Xiao Wang","email":"","orcid":"","institution":"East China University of Science and Technology","correspondingAuthor":false,"prefix":"","firstName":"Xiao","middleName":"","lastName":"Wang","suffix":""},{"id":449283423,"identity":"39bc5624-d491-4ad9-bf23-6bf268639ec9","order_by":6,"name":"Meng Zhang","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA2ElEQVRIiWNgGAWjYFACHhBhA+MxE60lDUQwNpCi5TAJWnT7zx78XPDrvBz/jPTnDxgqrBMb2M8ewKvF7MC5ZOmZfbeNJW7kGDYwnElPbODJS8Cv5WCPgTRvz+3EDRI5jA2MbYcTGyR4DPBrOcxj/Ju351z9Bon0hw2M/4jRcozHTJrnx4EEA4kEwwbGBmK0nOExs+ZtSDacceaN4YyEY+nGbTw5BLScP2N8m+ePnTx/e/qDDx9qrGX72c/g1wIGjG1QRgIQsxFWDwJ/iFM2CkbBKBgFIxQAANNIRVVLfJ3vAAAAAElFTkSuQmCC","orcid":"","institution":"East China University of Science and Technology","correspondingAuthor":true,"prefix":"","firstName":"Meng","middleName":"","lastName":"Zhang","suffix":""}],"badges":[],"createdAt":"2025-01-26 13:53:04","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-5906730/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-5906730/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":81627189,"identity":"96c2cbe6-59a3-48df-9309-aad97b02401c","added_by":"auto","created_at":"2025-04-29 10:38:22","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":278393,"visible":true,"origin":"","legend":"\u003cp\u003eIllustration of the anatomy of the Ni\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eS\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e clusters: (left) the shell S clusters, (middle) the core Ni clusters, (right) the assembled Ni\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eS\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e clusters. The angle represents rotation around the axis perpendicular to the screen.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-5906730/v1/458eb5e1476e3960afb92367.png"},{"id":81627793,"identity":"cf2a0886-3311-43a9-a060-0eb89cbbbd1d","added_by":"auto","created_at":"2025-04-29 10:46:23","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":175759,"visible":true,"origin":"","legend":"\u003cp\u003eThe growth model of Ni\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e core clusters for the lowest-energy structures of the Ni\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eS\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e clusters with the size evolution. The point group symmetries of these Ni clusters are C\u003csub\u003e1\u003c/sub\u003e, D\u003csub\u003e∞h\u003c/sub\u003e, D\u003csub\u003e3h\u003c/sub\u003e, D\u003csub\u003e2h\u003c/sub\u003e, C\u003csub\u003e4v\u003c/sub\u003e, O\u003csub\u003eh\u003c/sub\u003e, C\u003csub\u003e3v\u003c/sub\u003e, C\u003csub\u003e2v\u003c/sub\u003e, D\u003csub\u003e3h\u003c/sub\u003e, and C\u003csub\u003es\u003c/sub\u003e with tolerance 0.1Ǻ for \u003cem\u003en\u003c/em\u003e=1-10, respectively.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-5906730/v1/7dec6f2f1cbc574aa5a4df0a.png"},{"id":81627224,"identity":"165e989a-2015-42ed-8799-399fd97096ca","added_by":"auto","created_at":"2025-04-29 10:38:23","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":131318,"visible":true,"origin":"","legend":"\u003cp\u003eThe optimized unit cells of β- and α-NiS crystal structures (left) and their corresponding cluster structures from the top view and the side view (right).\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-5906730/v1/7cca5f4c69c77c2fad6fd4bf.png"},{"id":81627185,"identity":"05911584-41c5-4636-8714-03b8aaff4bb3","added_by":"auto","created_at":"2025-04-29 10:38:21","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":333293,"visible":true,"origin":"","legend":"\u003cp\u003eOptimized structures of the isolated Ni\u003csub\u003e3\u003c/sub\u003eS (a), F\u003csub\u003e6\u003c/sub\u003eS, Ni\u003csub\u003e6\u003c/sub\u003eS, Ni\u003csub\u003e6\u003c/sub\u003eSe, Ni\u003csub\u003e6\u003c/sub\u003eO, Zn\u003csub\u003e6\u003c/sub\u003eS (b), and Ni\u003csub\u003e3\u003c/sub\u003eS-H\u003csub\u003e3 \u003c/sub\u003e(c) clusters and the corresponding molecular orbital diagram. The isosurface value is 0.03 e/Ǻ\u003csup\u003e3\u003c/sup\u003e.\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-5906730/v1/2131b0f138e1ed350a2f1854.png"},{"id":81627794,"identity":"62ab0a46-4130-4dd4-a5b0-859600bf05e3","added_by":"auto","created_at":"2025-04-29 10:46:23","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":74364,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eE\u003c/em\u003e\u003csub\u003eb\u003c/sub\u003e (left) and Δ\u003csub\u003e2\u003c/sub\u003e\u003cem\u003eE \u003c/em\u003e(right) of the Ni\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eS\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e\u003csub\u003e \u003c/sub\u003eclusters (\u003cem\u003en\u003c/em\u003e=1-10) for the lowest-energy structures and cohesive energy of β-NiS bulk phase.\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-5906730/v1/d0d8ed943032f205cf9a8542.png"},{"id":81627799,"identity":"18f72b52-4b2c-474a-9a50-eef7b8915c19","added_by":"auto","created_at":"2025-04-29 10:46:25","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":120054,"visible":true,"origin":"","legend":"\u003cp\u003eRelative potential energy (eV) of the Ni\u003csub\u003e5\u003c/sub\u003eS\u003csub\u003e5\u003c/sub\u003e and Ni\u003csub\u003e6\u003c/sub\u003eS\u003csub\u003e6 \u003c/sub\u003eclusters for the lowest-energy structures during 20 ps of molecular dynamics simulation.\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-5906730/v1/02b5053cc42cdbd4296aed48.png"},{"id":81627169,"identity":"aca485b8-3003-4d46-bde9-d4cc120b30da","added_by":"auto","created_at":"2025-04-29 10:38:20","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":192697,"visible":true,"origin":"","legend":"\u003cp\u003eThe DED (a), HOMO, and LUMO orbitals (b), and PDOS (c) of the Ni\u003csub\u003e6\u003c/sub\u003eS\u003csub\u003e6 \u003c/sub\u003eand NiS cluster. Charge accumulations are obvious in purple regions. Green indicates that the wave function has a positive sign, while red indicates a negative sign. The surface isovalue for electron density is 0.03 e/Å\u003csup\u003e3\u003c/sup\u003e. The vertical dashed-dotted lines indicate the Fermi level.\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-5906730/v1/d24221ac37b7896c28826366.png"},{"id":81627243,"identity":"bcd9c29e-aa1a-4fe6-9f87-6c23b7b1bdda","added_by":"auto","created_at":"2025-04-29 10:38:24","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":73072,"visible":true,"origin":"","legend":"\u003cp\u003eThe average atomic charge on the S atom in a.u. (a), and the HOMO-LUMO energy gap (\u003cem\u003eE\u003c/em\u003e\u003csub\u003egap\u003c/sub\u003e) in eV (b) of the Ni\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eS\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e\u003csub\u003e \u003c/sub\u003eclusters (\u003cem\u003en\u003c/em\u003e=1-10) and β-NiS bulk phase for the lowest-energy structures.\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-5906730/v1/01d5f48ce672b80005e85d54.png"},{"id":81627798,"identity":"0d4215af-f3b0-4391-8508-dcde7746b422","added_by":"auto","created_at":"2025-04-29 10:46:25","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":411876,"visible":true,"origin":"","legend":"\u003cp\u003eOptimized structures of the Ni₄S₄ cluster adsorbing 1 to 16 hydrogen atoms. Hydrogen atoms are shown in white.\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-5906730/v1/f113473bbdf29e45bbc8b4cb.png"},{"id":95314509,"identity":"d865b69e-15cc-4aed-9bed-fa463e5e6b94","added_by":"auto","created_at":"2025-11-06 15:52:57","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3316909,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-5906730/v1/c3d3f78e-6b01-4fd0-bc2d-ef36407a8de9.pdf"},{"id":81627215,"identity":"178c0e66-ed46-4b85-9228-1f32e4a00ad5","added_by":"auto","created_at":"2025-04-29 10:38:23","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"supplement","size":989111,"visible":true,"origin":"","legend":"","description":"","filename":"Supportinginformation.pdf","url":"https://assets-eu.researchsquare.com/files/rs-5906730/v1/20b6c63cf6dfc8467a221457.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"\u003cp\u003e\u003cstrong\u003eStructural disassembly, dangling bond and hydrogen storage of Ni\u003c/strong\u003e\u003csub\u003e\u003cem\u003e\u003cstrong\u003en\u003c/strong\u003e\u003c/em\u003e\u003c/sub\u003e\u003cstrong\u003eS\u003c/strong\u003e\u003csub\u003e\u003cem\u003e\u003cstrong\u003en\u003c/strong\u003e\u003c/em\u003e\u003c/sub\u003e\u003cstrong\u003e (\u003c/strong\u003e\u003cem\u003e\u003cstrong\u003en\u003c/strong\u003e\u003c/em\u003e\u003cstrong\u003e=2-10) clusters\u003c/strong\u003e\u003c/p\u003e","fulltext":[{"header":"1. INTRODUCTION","content":"\u003cp\u003eEnvisaged as an intermediate between molecules to bulk solids, clusters usually exhibit distinct structures, physical and chemical characteristics from their atomic and condensed counterparts[\u003cspan additionalcitationids=\"CR2 CR3 CR4\" citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]. A critical concern in understanding these quasi-particle properties stems from geometrical determinations of the most stable isomers within the sub-nanometer to nanometer scales. Different from the isolated atoms or bulk phases, spatial arrangements of the clusters seldom follow certain patterns that can be generalized for cluster peers. Despite the latest progress in algorithms and computational powers, typical optimization procedures still undergo isomer constructions, enumerations, energy comparison, and stability investigations. Yet, structural evolution of these \u0026ldquo;building bricks\u0026rdquo; has remained a mystery since the emergence of cluster science from the 1980s[\u003cspan additionalcitationids=\"CR7 CR8\" citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eGeometric determinations are even more complicated in clusters containing few elements than the mono-element ones. Besides the aggregations of individual elements, the bonding between the elements brings additional factors during overall structural optimizations. A typical example can be retrieved from the transition-metal sulfide (TMS) clusters. Due to the bulks\u0026rsquo; importance in catalysis, combustion, and biological systems[\u003cspan additionalcitationids=\"CR11 CR12 CR13\" citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e], the TMS clusters have been extensively studied experimentally[\u003cspan additionalcitationids=\"CR16 CR17 CR18 CR19 CR20 CR21 CR22 CR23\" citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e] and theoretically[\u003cspan additionalcitationids=\"CR26 CR27 CR28 CR29\" citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e]. Moreover, the chalcogenides of simple stoichiometries, such as 1:1 and 1:2, are the most common and important TMS formula. Even though the TM\u003csub\u003e\u003cem\u003ex\u003c/em\u003e\u003c/sub\u003eS\u003csub\u003e\u003cem\u003ey\u003c/em\u003e\u003c/sub\u003e (TM\u0026thinsp;=\u0026thinsp;Mn, Fe, Co, Zn, and Cu, \u003cem\u003ex\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1, \u003cem\u003ey\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1 or 2) clusters have been well explored[\u003cspan additionalcitationids=\"CR27 CR28 CR29 CR30\" citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e], the individual researches could not end up with a structural evolution theorem despite composition, stability, structure, and photodissociation elucidations. The composed TM\u003csub\u003e\u003cem\u003ex\u003c/em\u003e\u003c/sub\u003eS\u003csub\u003e\u003cem\u003ey\u003c/em\u003e\u003c/sub\u003e clusters do not abide by any geometrical patterns of the TM\u003csub\u003e\u003cem\u003ex\u003c/em\u003e\u003c/sub\u003e or S\u003csub\u003e\u003cem\u003ey\u003c/em\u003e\u003c/sub\u003e clusters. Structural connections between the compound and its component clusters may be faint or unexplored. However, it has been noticed that many macro-clusters can be divided by motifs from individual components, and the metal cluster cores are covered by organic shells[\u003cspan additionalcitationids=\"CR33\" citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e34\u003c/span\u003e]. Thus, it is important to check if such a dissembling scheme also exists in small clusters such as these TMSs.\u003c/p\u003e \u003cp\u003eHydrogen is an efficient, clean, and cost-effective energy source due to its high energy density. In recent years, H₂ has gained significant attention as a promising alternative energy carrier. Meanwhile, low-cost transition metal-based nanostructures with various dimensions and morphologies (e.g., 0D, 1D, 2D, and 3D), such as transition metal chalcogenides, have been extensively explored for energy applications[\u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e35\u003c/span\u003e]. Among them, MoS₂ has been widely studied as a promising catalyst for the hydrogen evolution reaction (HER) due to its unique electronic structure and high catalytic activity at edge sites[\u003cspan additionalcitationids=\"CR37 CR38\" citationid=\"CR36\" class=\"CitationRef\"\u003e36\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e39\u003c/span\u003e]. Previous studies have shown that the incorporation of different elements can effectively modulate the catalytic properties of MoS₂, further enhancing its HER performance[\u003cspan additionalcitationids=\"CR41 CR42\" citationid=\"CR40\" class=\"CitationRef\"\u003e40\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e43\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eThe Ni\u003csub\u003e\u003cem\u003ex\u003c/em\u003e\u003c/sub\u003eS\u003csub\u003e\u003cem\u003ey\u003c/em\u003e\u003c/sub\u003e binary system is ideal for sorting out the shortcuts in growth patterns of the compound clusters. The NiS is dimorphic with a hexagonal phase α-NiS at high temperature[\u003cspan additionalcitationids=\"CR45 CR46\" citationid=\"CR44\" class=\"CitationRef\"\u003e44\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR47\" class=\"CitationRef\"\u003e47\u003c/span\u003e] and trigonal form β-NiS (millerite) at low temperature[\u003cspan additionalcitationids=\"CR49 CR50 CR51 CR52 CR53\" citationid=\"CR48\" class=\"CitationRef\"\u003e48\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR54\" class=\"CitationRef\"\u003e54\u003c/span\u003e]. Both positive and negative Ni\u003csub\u003e\u003cem\u003ex\u003c/em\u003e\u003c/sub\u003eS\u003csub\u003e\u003cem\u003ey\u003c/em\u003e\u003c/sub\u003e clusters were verified experimentally through the laser ablation of NiS isotope-labeled samples[\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e, \u003cspan citationid=\"CR55\" class=\"CitationRef\"\u003e55\u003c/span\u003e]. The negative particles are constituted of clusters with similar fractions of Ni and S atoms. Besides, Ni and S formed clusters with a larger content of S in NiS\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e (\u003cem\u003en\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1\u0026ndash;4)[\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e] and preferably bonded at squeezed spaces[\u003cspan citationid=\"CR56\" class=\"CitationRef\"\u003e56\u003c/span\u003e]. The NiS also acted as a nanocrystal catalyst[\u003cspan citationid=\"CR57\" class=\"CitationRef\"\u003e57\u003c/span\u003e] or shielding component to cage clusters for efficient hydrogen evolution under different circumstances[\u003cspan citationid=\"CR58\" class=\"CitationRef\"\u003e58\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eIn this paper, we present a systematic investigation of Ni\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eS\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e (\u003cem\u003en\u003c/em\u003e\u0026thinsp;=\u0026thinsp;2\u0026ndash;10) cluster properties, growth patterns, and their structural correlations to elementary Ni and S clusters by using density functional theory (DFT) calculations and particle swarm optimization. The calculated results demonstrate that Ni\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eS\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e clusters can be anatomized to Ni\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e and S\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e clusters with core/shell self-similarity morphology, which can be extended to NiS millerite and TMS crystal structure. The predicted NiS clusters own a large number of dangling bonds that facilitate the absorbance of hydrogen. This work provides a possible shortcut in structural optimizations for functional clusters by assembling or dissembling the compound clusters.\u003c/p\u003e"},{"header":"2. COMPUTATIONAL DETAILS","content":"\u003cp\u003eInitial structures of Ni\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eS\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e clusters were generated using the CALYPSO software which has been verified highly efficient for cluster structures prediction[\u003cspan citationid=\"CR59\" class=\"CitationRef\"\u003e59\u003c/span\u003e, \u003cspan citationid=\"CR60\" class=\"CitationRef\"\u003e60\u003c/span\u003e]. A local version of the particle swarm optimization (PSO) algorithm was implemented to explore the potential energy surface for non-periodic systems[\u003cspan citationid=\"CR61\" class=\"CitationRef\"\u003e61\u003c/span\u003e]. In this work, Gaussian09 code[\u003cspan citationid=\"CR62\" class=\"CitationRef\"\u003e62\u003c/span\u003e] with PBEPBE/LANL2DZ[\u003cspan citationid=\"CR63\" class=\"CitationRef\"\u003e63\u003c/span\u003e, \u003cspan citationid=\"CR64\" class=\"CitationRef\"\u003e64\u003c/span\u003e] basis set was employed in local geometry optimization and total energy calculations. More than 2000 distinct isomers were selected from the CALYPSO results for finer optimization by using DFT implemented in the DMOL\u003csup\u003e3\u003c/sup\u003e program[\u003cspan citationid=\"CR65\" class=\"CitationRef\"\u003e65\u003c/span\u003e]. All-electron spin-unrestricted calculations were performed with a double numerical precision plus polarization with the addition of diffuse functions (DNP+). Relativistic calculations were carried out with scalar relativistic corrections to valence orbitals relevant to atomic bonding properties via a local pseudopotential with the Perdew -Burke-Ernzerhof (PBE) functional[\u003cspan citationid=\"CR66\" class=\"CitationRef\"\u003e66\u003c/span\u003e]. Convergence of self-consistent field (SCF) was set with a criterion of 1\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;5\u003c/sup\u003e Hartree on total energy and electron density, 2\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003e Hartree/\u0026Aring; on the gradient, and 5\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003e\u0026Aring; on the displacement, respectively. Harmonic vibrational analysis was carried out at the same level of theory to examine low-energy isomer stabilities and estimate zero-point energy corrections. To verify computational reliability, we performed a benchmark calculation on NiS dimer using different type of the gradient-corrected hybrid BP[\u003cspan citationid=\"CR67\" class=\"CitationRef\"\u003e67\u003c/span\u003e], BLYP[\u003cspan citationid=\"CR68\" class=\"CitationRef\"\u003e68\u003c/span\u003e], PW91[\u003cspan citationid=\"CR69\" class=\"CitationRef\"\u003e69\u003c/span\u003e] and PBE functionals in DMOL\u003csup\u003e3\u003c/sup\u003e for bond length and vibrational frequency. The results are listed in Table \u003cspan refid=\"MOESM1\" class=\"InternalRef\"\u003eS1\u003c/span\u003e, where the PBE method shows the best agreement with experimental values[\u003cspan additionalcitationids=\"CR71\" citationid=\"CR70\" class=\"CitationRef\"\u003e70\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR72\" class=\"CitationRef\"\u003e72\u003c/span\u003e] and hence we selected the functional PBE in this study.\u003c/p\u003e"},{"header":"3. RESULTS AND DISCUSSION","content":"\u003ch2\u003e3.1. Individual Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e Cluster Study\u003c/h2\u003e\n\u003cp\u003eVarious low-lying isomers of Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters (\u003cem\u003en\u003c/em\u003e=1-10) are determined with global structure search and DFT calculations.\u0026nbsp;The ground-state structures\u0026nbsp;of\u0026nbsp;Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e isomers (\u003cem\u003en\u003c/em\u003e up to 10) are presented in Figure 1. Other metastable isomers are listed in Figure S1 and Table S2 in the Supporting Information, in the order of relative energies and structural coordinates. Schematic representation in Figure 1 illustrates the anatomy of the lowest-energy structures of Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters.\u003c/p\u003e\n\u003cp\u003eThe diatomic NiS has a C\u003csub\u003e\u0026infin;\u003c/sub\u003e\u003csub\u003ev\u003c/sub\u003e symmetry and a bond length of 1.986 \u0026Aring;. The planar rhombus with Ni-S bond length of 2.065 \u0026Aring; is the ground-state geometry for Ni\u003csub\u003e2\u003c/sub\u003eS\u003csub\u003e2\u003c/sub\u003e cluster that can be seen as a crisscross of Ni\u003csub\u003e2\u003c/sub\u003e and S\u003csub\u003e2\u003c/sub\u003e dimers. The Ni\u003csub\u003e3\u003c/sub\u003eS\u003csub\u003e3\u003c/sub\u003e has a quasi-planar equilateral triangle structure with a planar Ni\u003csub\u003e3\u003c/sub\u003e triangle coated by a S\u003csub\u003e3\u003c/sub\u003e triangle. Each bridging S atom is bonded with two Ni atoms in the cluster. The dimensional enhancement from 2D \u0026rarr; 3D starts at \u003cem\u003en\u003c/em\u003e = 4 as shown in Figure 1. The square antiprism structure is the most stable structure for Ni\u003csub\u003e4\u003c/sub\u003eS\u003csub\u003e4\u003c/sub\u003e. The Ni\u003csub\u003e4\u003c/sub\u003e tetrahedron is embedded in the S\u003csub\u003e4\u003c/sub\u003e tetrahedron and each S atom is bonded to three Ni atoms through a face-capped model. Ni\u003csub\u003e5\u003c/sub\u003eS\u003csub\u003e5\u003c/sub\u003e cluster can be regarded as docking the square pyramids of Ni\u003csub\u003e5\u003c/sub\u003e and S\u003csub\u003e5\u003c/sub\u003e cluster with four Ni\u003csub\u003e3\u003c/sub\u003eS faces capping triangular.\u0026nbsp;The most stable\u0026nbsp;structure of\u0026nbsp;Ni\u003csub\u003e6\u003c/sub\u003eS\u003csub\u003e6\u003c/sub\u003e can be generated by stacking the two Ni\u003csub\u003e3\u003c/sub\u003eS\u003csub\u003e3\u003c/sub\u003e triangles. Both the core Ni\u003csub\u003e6\u003c/sub\u003e and shell S\u003csub\u003e6\u003c/sub\u003e clusters are octahedrons with each S atom face-capping the Ni\u003csub\u003e3\u003c/sub\u003e triangle. From the cluster size of \u003cem\u003en\u003c/em\u003e=7, the Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e cluster can be evolved by successively capping the N\u003csub\u003e6\u003c/sub\u003eS\u003csub\u003e6\u003c/sub\u003e. As shown later, the structures of Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e and S\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e (\u003cem\u003en\u003c/em\u003e=7-10) can be considered as mono-face-capped, bi-capped, tri-capped, and tetra-capped octahedral structures.\u003c/p\u003e\n\u003cp\u003eA close look towards cluster geometries denotes 1) the compound clusters own self-similarity and nested-assembly characteristics to their component peers, and 2) a building-brick of an S atom face-capping a triangular Ni\u003csub\u003e3\u003c/sub\u003e turns out in all 3D clusters from \u003cem\u003en\u003c/em\u003e=4 to \u003cem\u003en\u003c/em\u003e=10. The former behavior indicates that the geometries and structural evolution of Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters may be relevant to these of the mono-compositional clusters, and the latter shows Ni\u003csub\u003e3\u003c/sub\u003eS tetrahedral configuration may act as a seed to drive cluster development.\u003c/p\u003e\n\u003ch2\u003e3.2. Cluster Disassembly and Assembly\u003c/h2\u003e\n\u003cp\u003eThe possibility of an assembly-disassembly scheme in the Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters is studied. Two important features towards the geometries of the Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters are noticed in Figure 1. (1) All the Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters have core-shell structures where the nickel stays as core and sulfur seats on shells. Nickel atoms stacked together, nucleated, and then subsequently bonded to sulfur atoms. The resulting Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters were stretched to three-dimensional (3D) composites when \u003cem\u003en\u003c/em\u003e\u0026ge;4. Each S atom favors\u0026nbsp;face-capping on the triangular\u0026nbsp;Ni plane via bonding with three Ni atoms in all 3D structures. On the contrary,\u0026nbsp;distinct segregation between the\u0026nbsp;sulfur\u0026nbsp;atoms is observed\u0026nbsp;due to the absence of S-S bonds. (2) The Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters can be viewed as assemblies of individual Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e and S\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters whose atoms keep the same spatial orientations as if they were in the binary systems. One \u0026ldquo;matryoshka doll\u0026rdquo; Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e cluster owns a specifically oriented sulfur shell, which is filled by its nickel miniature. In another word, peeling away the S atoms emerges the Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e cluster whose structure is similar to the S\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e cluster. To assemble the Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e, the low-lying S\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e and Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters turn certain angles against and then inlay with each other to maximize spatial occupancies. In most cases, sulfur clusters are inversely placed to nickel ones as if the formers are turned 180 degrees or vice versa. Exceptions were found in Ni\u003csub\u003e2\u003c/sub\u003eS\u003csub\u003e2\u003c/sub\u003e and Ni\u003csub\u003e8\u003c/sub\u003eS\u003csub\u003e8\u003c/sub\u003e clusters, where 90 and 45 degrees clockwise rotations are applied respectively. Ni\u003csub\u003e7\u003c/sub\u003eS\u003csub\u003e7\u003c/sub\u003e cluster has a chiral structure as shown in Figure 1, which was predicted in bimetallic and trimetallic alloy clusters by doping hetero metal atoms to lower cluster symmetries[73,74]. Based on the above knowledge, Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e are nested assemblies from the individual metallic and nonmetallic clusters.\u003c/p\u003e\n\u003cp\u003eThe individual Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e and S\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters have the same point group at a corresponding number \u003cem\u003en\u003c/em\u003e. The core Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e cluster turns to the shell S\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters when Ni atoms are replaced with sulfur atoms. Sulfur shell undergoes a self-similarity process from the Ni core. These small clusters can be viewed as inorganic self-similarity cases in the formation of geometric structures similar to the well-known double helix of DNA, which has not been reported in TMS clusters of MnS, FeS, and CoS [18,75]. We also note that the assembly-disassembly mechanism has been reported for single element clusters. Experimentally the supported Ag nanoclusters followed a construction pattern that can be separated for individual Ag clusters[76]. Similar behavior has also been observed in Si clusters. Previous DFT calculation and global minimum search have shown that the low-lying isomers of Si clusters can be derived from assembling tricapped trigonal prisms [77].\u003c/p\u003e\n\u003cp\u003eThanks to the assembly-disassembly scheme, investigation of the growth patterns of the compound Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e cluster is simplified to study its component Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e or S\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e cluster. Here we take the Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e cluster as an example and visualize its growth pattern in Figure 2. A successive Ni\u003cem\u003e\u003csub\u003en+\u003c/sub\u003e\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e cluster can be generally evolved from a Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e\u003csub\u003e\u0026nbsp;\u003c/sub\u003ecluster by capping a Ni atom. Ni\u003csub\u003e6\u003c/sub\u003e is a 3D octahedral structure which is the basic building block for the core Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e cluster from the size of \u003cem\u003en\u003c/em\u003e=6. Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e (\u003cem\u003en\u003c/em\u003e=7-10) clusters and can be viewed as mono-, bi-, tri- and tetra-capped octahedral structures.\u003c/p\u003e\n\u003cp\u003eThe peeled Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e core clusters from the Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e system obtained in this work differ from those of small bare nickel clusters in the gas phase. Previous DFT studies have predicted that pure Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters in the intermediate size range (\u003cem\u003en\u003c/em\u003e =6-10) prefer a pentagonal pyramid growth pattern[78-82]. This infers that the sulfur atoms exert a tremendous influence on the structures of core Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters when S atoms interact with the Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters. Since the Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e cluster has similar structures as the S\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e cluster, the shell S\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e cluster also has the atom-by-atom growth principle which is also different from the pure S\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e cases[83].\u003c/p\u003e\n\u003cp\u003eThe formation pattern studies of Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters elucidate growth processes from NiS cluster to the crystal. Crystalline NiS is dimorphic. The \u0026beta;-NiS millerite structure with trigonal \u003cem\u003eR\u003csub\u003e3m\u003c/sub\u003e\u003c/em\u003e is found as the stable crystalline phase here, consistent with previous experimental and theoretical results[48-54]. Interestingly, the crystalline \u0026beta;-NiS and its corresponding cluster counterparts have the common core-shell structure as shown in Figure 3. The unit cell of \u0026beta;-NiS displays three S atoms sitting around a Ni\u003csub\u003e3\u0026nbsp;\u003c/sub\u003etriangle. Besides, the self-similarity character and nested-assembly process are also seen in the cluster structures of crystalline NiS. The average bond lengths are also calculated for the binary clusters and shown in Figure S2 along with the shortest Ni-S bond length. It is found that the average bond length increases monotonously with the number of atoms and gradually approaches the bond length of the \u0026beta;-NiS bulk phase. At \u003cem\u003en\u003c/em\u003e =10, the average Ni-S bond length of Ni\u003csub\u003e10\u003c/sub\u003eS\u003csub\u003e10\u003c/sub\u003e cluster is 2.22 \u0026Aring;, very close to 2.28 \u0026Aring;, in the \u0026beta;-NiS crystal.\u003c/p\u003e\n\u003cp\u003eIt is worth mentioning that in general, clusters have little or no common structural features with the bulk phases. However, the present work gives strong evidence of the similarity between the small gaseous Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters and NiS bulk. The finding of the characteristics shared by the clusters and their bulk provides important insights towards the size-evolution mechanism of the geometrical structures from atom to cluster to the condensed state. This is very crucial in the discovery of novel nanomaterials in transition-metal sulfides series.\u003c/p\u003e\n\u003ch2\u003e3.3. Physical Prerequisites for Assembly-Disassembly Structures\u003c/h2\u003e\n\u003cp\u003eThe assembly-disassembly scheme does not apply to all clusters. The physical prerequisites for such a mechanism deserve a thorough investigation to simplify future functional cluster predictions. In this subsection, we figure out the requirements of such a scheme through elucidations of molecular orbital hybridizations in Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters.\u003c/p\u003e\n\u003cp\u003eWe start with an exploration of the seeding structure of the Ni\u003csub\u003e3\u003c/sub\u003eS. A building block from the Ni\u003csub\u003e3\u003c/sub\u003eS is noticed in 3D Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters as shown in Figure. 1. Each trigonal pyramid has three Ni-S bonds with the S-capped Ni\u003csub\u003e3\u003c/sub\u003e triangles. Maximizing the coordination number of Ni-S and Ni-Ni bonds will lead to the high stability of the Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters. A molecular orbital (MO) analysis of the bare Ni\u003csub\u003e3\u003c/sub\u003eS cluster was performed. Figure 4 (a) gives the isolated Ni\u003csub\u003e3\u003c/sub\u003eS cluster and its occupied molecular orbital and it is evident from the MO of Ni\u003csub\u003e3\u003c/sub\u003eS that the S atom bonds to each Ni. Three d\u003csub\u003ez\u003c/sub\u003e orbitals of Ni atoms are overlapping with the orbitals of S atom, yielding three \u0026sigma; bonds featured in MO.\u003c/p\u003e\n\u003cp\u003eThe orbital stretching of bare Ni\u003csub\u003e3\u003c/sub\u003eS varies in the Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e systems subjected to additional bonds to deficient or excessive atoms. Table 1 tabulates charge contributions from 3s, 3p, and 3d atomic orbitals on each S atom in Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters with Mulliken Population analysis. A significant amount of charge is retained in the 3d orbital of S atom. The charge occupations in 3d orbital of the Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters at \u003cem\u003en\u003c/em\u003e\u0026ge;4 are significantly larger than these of the planar structure at \u003cem\u003en\u003c/em\u003e\u0026le;3. Noteworthy, the planarly-structured Ni\u003csub\u003e1\u003c/sub\u003eS\u003csub\u003e1\u003c/sub\u003e, Ni\u003csub\u003e2\u003c/sub\u003eS\u003csub\u003e2\u003c/sub\u003e, and Ni\u003csub\u003e3\u003c/sub\u003eS\u003csub\u003e3\u0026nbsp;\u003c/sub\u003eclusters cannot form the structure of the Ni\u003csub\u003e3\u003c/sub\u003eS trigonal pyramid, which is the characteristic structural feature of 3D Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters. Combining the charge distribution and structure characteristics, we infer that the S atom has a certain sp\u003csup\u003e3\u003c/sup\u003ed\u003csup\u003e2\u003c/sup\u003e type hybridization involving 3s, 3p, and 3d orbitals in the 3D Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters[84-86]. It is similar to the case in an isolated F\u003csub\u003e6\u003c/sub\u003eS molecule where the \u003cem\u003ed\u003c/em\u003e orbitals from sulfur are well known to participate in the hybridization[87,88]. Such an orbital hybridization scheme is in charge of spatial stretching of the clusters that lead to the assembly-disassembly feature eventually.\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;\u003cstrong\u003eTable\u0026nbsp;\u003c/strong\u003e\u003cstrong\u003e1\u003c/strong\u003e Charge distribution in 3s, 3p, and 3d orbitals of per S atom of the Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e system from Mulliken Population analysis.\u003c/p\u003e\n\u003cdiv align=\"\"\u003e\n \u003ctable border=\"0\" cellspacing=\"0\" cellpadding=\"0\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd rowspan=\"2\" style=\"width: 68px;\"\u003e\n \u003cp\u003eSystems\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd colspan=\"3\" style=\"width: 204px;\"\u003e\n \u003cp\u003eOrbitals\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e3s\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e3p\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e3d\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003eNi\u003csub\u003e1\u003c/sub\u003eS\u003csub\u003e1\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e1.890e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e4.221e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e0.051e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003eNi\u003csub\u003e2\u003c/sub\u003eS\u003csub\u003e2\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e1.834e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e4.212e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e0.100e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003eNi\u003csub\u003e3\u003c/sub\u003eS\u003csub\u003e3\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e1.835e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e4.213e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e0.088e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003eNi\u003csub\u003e4\u003c/sub\u003eS\u003csub\u003e4\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e1.811e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e4.123e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e0.176e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003eNi\u003csub\u003e5\u003c/sub\u003eS\u003csub\u003e5\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e1.812e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e4.102e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e0.176e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003eNi\u003csub\u003e6\u003c/sub\u003eS\u003csub\u003e6\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e1.812e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e4.101e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e0.171e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003eNi\u003csub\u003e7\u003c/sub\u003eS\u003csub\u003e7\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e1.812e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e4.096e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e0.177e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003eNi\u003csub\u003e8\u003c/sub\u003eS\u003csub\u003e8\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e1.814e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e4.022e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e0.178e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003eNi\u003csub\u003e9\u003c/sub\u003eS\u003csub\u003e9\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e1.805e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e4.034e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e0.177e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003eNi\u003csub\u003e10\u003c/sub\u003eS\u003csub\u003e10\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e1.802e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e4.001e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e0.179e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003eNi\u003csub\u003e3\u003c/sub\u003eS\u003csub\u003e1\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e1.773e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e4.104e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e0.190e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003eNi\u003csub\u003e6\u003c/sub\u003eS\u003csub\u003e1\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e1.697e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e3.446e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e0.524e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eTo verify this hypothesis, we optimized the structure of the F\u003csub\u003e6\u003c/sub\u003eS cluster and compared it with other Ni\u003csub\u003e6\u003c/sub\u003eX (X refers to the chalcogenide element of O, S, Se) and the Zn\u003csub\u003e6\u003c/sub\u003eS clusters. Different elements of chalcogen were selected to have an expanded view of the sp\u003csup\u003e3\u003c/sup\u003ed\u003csup\u003e2\u003c/sup\u003e type hybridization in these similar clusters. The Zn atom is considered to analyses the effects of the interactions and bonding modes with various \u003cem\u003ed\u003c/em\u003e orbitals in the TM participants compared with Ni. The geometric structures and corresponding MOs are depicted in Figure. 4(b). The octahedral F\u003csub\u003e6\u003c/sub\u003eS has six fluorine atoms surrounding the central sulfur atom, while the Ni\u003csub\u003e6\u003c/sub\u003eS is a trigonal prismatic cluster with six nickel atoms surrounding the central sulfur atom. This is rational because there is an insufficient sp\u003csup\u003e3\u003c/sup\u003ed\u003csup\u003e2\u003c/sup\u003e orbital hybridization of the S atom in the Ni\u003csub\u003e6\u003c/sub\u003eS cluster. As a result, the octahedral geometry is distorted and displays low symmetry. In F\u003csub\u003e6\u003c/sub\u003eS, six sp\u003csup\u003e3\u003c/sup\u003ed\u003csup\u003e2\u003c/sup\u003e hybrid bond orbitals on the sulfur are overlapped by six fluorine 2p orbitals, yielding six \u0026sigma; (sigma) bonds. The MOs of Ni\u003csub\u003e6\u003c/sub\u003eS show that there are six equal Ni-S bonds with equal strength and length. The sulfur atom forms covalent S-Ni bonds with six surrounding nickels by d\u003csub\u003ez\u003c/sub\u003e-sp\u003csup\u003e3\u003c/sup\u003ed\u003csup\u003e2\u003c/sup\u003e overlap as shown in Figure 4(b). This indicates overlapping between Ni atom and S effectively yields Ni\u003csub\u003e6\u003c/sub\u003eS cluster with trigonal prismatic structure. The overall stability of the system is enhanced. These results are consistent with our previous discussions of the structures and stabilities of Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters. According to our structural postulate and analysis, Ni\u003csub\u003e6\u003c/sub\u003eSe will also exhibit the same structure and MO graph since Se and S have similar electronic configurations, in agreement with our MO calculations.\u003c/p\u003e\n\u003cp\u003eHowever, such \u0026sigma; d\u003csub\u003ez\u003c/sub\u003e-sp\u003csup\u003e3\u003c/sup\u003ed\u003csup\u003e2\u003c/sup\u003e overlap scheme is not suitable for the Zn\u003csub\u003e6\u003c/sub\u003eS case because d orbital is fully occupied by the valence electrons in IB and IIB groups and nearly no empty d orbitals can be provided for forming covalent Zn-S bonds. As clearly seen in the Zn\u003csub\u003e6\u003c/sub\u003eS, s orbitals in Zn participate in \u0026sigma; s\u0026ndash;sp\u003csup\u003e3\u003c/sup\u003ed\u003csup\u003e2\u003c/sup\u003e overlap, which is different from the Ni\u003csub\u003e6\u003c/sub\u003eX cases where the d\u003csub\u003ez\u003c/sub\u003e orbitals of Ni bond to the X elements. On the other hand, the trigonal prismatic structure of Ni\u003csub\u003e6\u003c/sub\u003eO failed to reach structural optimization due to the large energy difference between the relevant 2p (filled) and 3d (empty) orbitals in the oxygen atom. Consequentially, the outermost 2p electron can hardly jump to the 3d orbital to achieve sp\u003csup\u003e3\u003c/sup\u003ed\u003csup\u003e2\u003c/sup\u003e hybridization, which leads to the absent trigonal prismatic structure in the Ni\u003csub\u003e6\u003c/sub\u003eO cluster.\u003c/p\u003e\n\u003cp\u003eFrom the results, it was concluded that the disassembly and assembly core-shell structural motif might be attributed to the certain sp\u003csup\u003e3\u003c/sup\u003ed\u003csup\u003e2\u003c/sup\u003e type hybridization in the trigonal pyramid structure Ni\u003csub\u003e3\u003c/sub\u003eS, which is the principal structural unit in the formation of Doll-like structures of Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters. Such a hybrid mechanism may be useful in identifying new nanomaterials with binary self-similar structures.\u003c/p\u003e\n\u003ch2\u003e3.4. Stability Assessment\u003c/h2\u003e\n\u003cp\u003eThe stability of clusters is evaluated by the average binding energy (\u003cem\u003eE\u003c/em\u003e\u003csub\u003eb\u003c/sub\u003e) per atom and the second-order energy difference (\u0026Delta;\u003csub\u003e2\u003c/sub\u003e\u003cem\u003eE\u003c/em\u003e)\u0026nbsp;of the\u0026nbsp;Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u0026nbsp;\u003c/sub\u003e\u003c/em\u003eclusters\u003cem\u003e\u003csub\u003e\u0026nbsp;\u003c/sub\u003e\u003c/em\u003eand\u003cem\u003e\u003csub\u003e\u0026nbsp;\u003c/sub\u003e\u003c/em\u003e\u0026beta;-NiS bulk phase. The corresponding formulas are given below:\u003c/p\u003e\n\u003cp\u003e\u003cimg 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AABoApIjAAAAB8kRAACAg+QIAADAQXIEAADgIDkCAABwkBwBAAA4SI4AAAAcJEcAAAAOkiMAAAAHyREAAICD5AgAAMBBcgQAAOAgOQIAAHCQHAEAADhIjgAAABwkRwAAAA6SIwAAAAfJEQAAgIPkCAAAwEFyBAAA4CA5AgAAcJAcAQAAOEiOAAAAHCRHAAAADpIjAAAAB8kRAACAg+QIAADAQXIEAADgIDkCAABwkBwBAAA4SI4AAAAcJEcAAAAOkiMAAAAHyREAAICD5AgAAMBBcgQAAOAgOQIAAHCQHAEAADhIjgAAABwkRwAAAA6SIwAAAAfJEQAAgIPkCAAAwEFyBAAA4CA5AgAAcJAcAQAAOEiOAAAAHCRHAAAADpIjAAAAB8kRAACAg+QIAADAQXIEAADgIDkCAABwkBwBAAA4SI4AAAAcJEcAAAAOkiMAAAAHyREAAICD5AgAAMBBcgQAAODobHL03nvvBXfddVfw9ddfh1O6Tet58803B5999lk4pX5vvPFGsOOOOwZz5syZft14443hX6szanXpwzDipUrERPXaHhN5jFL8pNVrXBv+xBNPhH8dbZ1MjhT4q1evDpYsWRJst9124dRu03ouW7YsuOmmm8z6+7R06dKg1+uZ16WXXhpOrcYo1qUPw4yXsoiJerQ5JvIYtfhJq9dDDjkk2LRpk2m7Jycnw6mQziVHyo6vvPLKYGJiYuQaTq3v6aefHlx88cWdOPob5br0oY3xQkzUq2ttSNSoxk/X67UOnUuOLr/88mBsbCzYeeedwymjRUcCxxxzjCmHthv1uvShbfFCTNSvS21I1CjHT9Pr9cknnwweeOCB8FM2H374YfD73/8++OKLL8Ip1elUcqTzp6+88kpw9NFHh1OqddZZZ5nzs/qdJtP6qxyavpxpqEt/2hIvxIQ/XWhDouqMH/YN5fzmN78J/vvf/wannnpqOCUwCY+mf/e73zVjofbaa69ZydO8efOCc845J7jooosqT5BKJ0f//Oc/Kz0/rS6/Pffcc8YAsaRXdODYLbfcEuy+++7B3nvvHU4JzHfc/6N5J3UrRr+rl4K+Dhokp/E5djCc3jVoTtPLDhTU+qscVB7DNCp1WUSd9Z+Xz3ghJuI1KR7ER0xov3HooYdO14X+PWinXWX8NDV26owFn9t6VnfccYd5dxMjOeOMM4K//vWvpqfrggsuCP7zn/8Ep512mkmYXHPnzg2uuOKK4Cc/+Um1CVKvpKVLl/YmJyfDT9WZmJjoafE0f9fmzZvN3/oF0tuwYUM4tdfbuHFjr59FJi7L+vXre2NjY2ae4+PjZj5JVq5cab777rvvhlOqpd/WMmh57W/Y9cqyfFmoHDR/lUtdVP6qh2gdRXW5LovwUf95+YgXFzGxVRPjQeqMCdt2aP3cl6Y9/vjj4beSVRk/7Bu20HT9Rpbyr8rU1JT5zQ8++CCcssULL7zQ6ydE4act7Hfjvi/6/pIlS8JP5ZVKjmyA17EBpVWUgmPZsmUzflPfG1Sxdp56pe3QNY9BO/wy0tZNf6tiA8hSHmXZ+h9UVl2uyyJ81H9eWcq8SsTEVk2MB8lS5kVoXbROSjTseikRyJqgSNXx05TY8RELSWWR9tt1WbhwYWxCc/vtt4f/mkkJkJZRyVOUpiX9rYhSp9XUNaduLA2Kev3118Op5anb8KmnnjLnE+fPnx9Onemwww6bMajuoYceMt1ru+66azglni7f1PfuvffeWu7Jk8Vbb72VuKy6okDrXZbmrd9QuQxT1+uyCB/1n5fPeCEmZmpiPEhdMaHTaTqFcvXVV09fMabTPY899phZ16mpqdShGnXFD/sGv958883gmWeeCX7wgx+EU7Y6//zzw3/N9L3vfc+8b7/99ubddcQRR5j3+++/37yXVTg50rnZbbbZJlizZo35XGVB9zPj4KOPPgr22GOPGYWwbt0687vaoDQIy9LG8umnn4af0vWPToK7777b/Pv666/PPDBNG+t5550X7LvvvpUMZlNSedlll5lld2mD1oZdFZVL9Dd8GoW6LMJX/eeVNV608yg65oKYmK2p8SBVtyG6aurcc88NP22ldbU7uDR1xQ/7Br+UGMmPfvQj857FV199ZcaGHXTQQeGUmRYsWJD7irckhZMjZWcKhuOPP95ks+vXr6+sUVEvlHqjjjvuuOkjC1XiO++8E5sx2o1lhx12CHbaaadwarITTjjB3LhQQXjyySebDSqN1uvwww83iaD+T1mXXHKJydyfe+45c/WALmF0acO2612UykHloXJR+QxL1+uyCB/1n5fPeCEmZmpiPMiw2pBB9Vxn/LBv8Of5558P/5Xdgw8+GPz2t78NP82msvvyyy+DF198MZxSXKHkSAHz2muvmW4+m+0rMIqsbBwFuSj5EgW+jlTff//9yhqJ2267LRgfHzcb2Zlnnml+I4mOdD7++GPz/SpofrfeequpSJWjEkwdeaR1JRelTPvzzz8PP/nX9boswmf95+UjXoiJmZocD+KrDdF+RTu1Cy+80OxXktQdP+wb/Npvv/3Cf6XTVW3qGYpe1eaq8hR0oeRImbtOqdkAttnun/70p4GZ9iAKRJ1PlhNPPDHQpYw6Gli5cmVwyimnmOlRqmBVdB7aiO677z5TmMrSdZ8En1TBL7/8sunKFR157LPPPuYSzrSNsQpqSNxLUtNe+m5Ro1KXRQyz/oeJmIg3qvHg0n5FdNCdxEf8jPK+wSd7Wk25wyAan/TII4+YxDWNHZNUhdzJkSpHgaOEyNJguoMPPthk2jbAi1KGrAF5ysTV5dfr9cz74sWLUwfTFaHk7uGHHzaVM4xBeCq3V199NVi7du10gOj5NyeddFKtG4E2MpVrlpe+W9Qo1WURw6r/YSImko1iPFg6qNaOT/WV1mvkK35Gdd/g08KFC8278oY0Oiu1YsUK0/liy8KH3MmRgnO33XYz3X+WMm2d/5WyA7M1SEyF4Z5P1vv++++feL7YnkMtQuthB+EtX7581o3DfNCRgrpmJyYmzGcdrdgbY7XZKNZlEU2vf+0YbE+ifal8tdOITh80SJuYGKyN8ZD0yppUTE5Omh4ad78Sx2f8sG/wY+PGjeG/ZlNd62aQq1atynTKrKqhPZI7OdLl+z/+8Y/DT1vZywzLDsy2l1xqMJpLCVncYLsqaBCeNk45++yzSy1/Fnfeeees39AGrobELoe6jqs4QlCjkNRo1G0U6rIIn/WfV1y8xPU0ajk1cDU6/Z577gn/VzxiYrYmx4NEY6Lqnmet5y677GLqaRDf8cO+oT7ROoxyE6Po1WmDEsQsVz0Okis5UqV98803wVFHHRVO2UrdkGUHZtsBeXGVZkfpKyiuu+66GcGhjUKXdZYZYKb7KqirVst/1VVXhVProeXUEVCcqu9rEr3cVbShxR3lxb303SJGpS6L8Fn/ecXFS1WIiXhNjgepMyZ02bXiIUsSNaz4GaV9g0+6GlBeeukl8+5SWetxIN/+9rfN2CQ9XNa+NGBfY5Di6Lt61EgVciVHSnr0/BsFYRw7IC5uYLaCVU8E1g5Xz4qJy8DtwDklWUnnnXXjsG233XbGMujfZc83ax52EJ4P6oFzN2DLloHbbVyEnY/KJTqfqo/84oxSXbqUTGa5/0/d9e/Kskxp8VKVUYyJtsaD1B0TSox0RVb0nkcauqErtKLlMaz48R07w9w3+KTy1LijuM4UJUb/+te/zKX7Gm/kvv785z/H9jLahElJZBUyJ0dKdhTIaT+sHiVl2HEDs7Uh/OEPfzA7XF2eGL3Jlf6tacoYDzzwwHDqFvqb7vegxEzd9nHdcUrM9H+Tsm7NQ3cfTeuS1AZnB+FFadCf7guh37CXkpahc8daD/c+FmoUdJSiwe16L0PloGVNuoKjTqNWl5Z2OhqbkEXd9W9lXaa642UUY6LN8SB1xoT2B3qIqAYZR3urdWWWLtN3d9x1xk/TYqfL+4YonTJTb090ULZyjeiBuvtatGhR+M2t1q1bZ+5yruS5Ev0fymTSefZMltfSlOfP6JlcY2Nj08++0Xfj5hH36idfsc+W0bz6mahZzqi4ZU9bvujzc+Iekhj3O1mtWbPGzFPPE9IzgOw89Rt6wGDc+uWl5VN5uM8Xqlrcs9VGoS61zNdee23ssmteacsjddR/2WXSd/LES5Z5WvqeW95pry5s364s5eSjPXBlrTt9r442ZO3atTPKOvrSeqs8rDrjR5+j8xhm7Axz32DLwuez1UTx0E9qwk/F6EG0CxYs6G3atCmcUl7m5KhKqvzFixdXvtEraJM2jjrZDc9u1Kps/TtLA5RFkfnrb1X9fhLbMNTxO5pnU+tSy1QmEckja92XXSb9vcrlroOWz3dMlN22q44H8bVM+k7Vyz5MWpe2xc8gReafVK/DSo5Ey61XEUqI9EDaqampcEo1cl+tVgV166WNXSpK91765JNPTBekT7bLtb/hTf/2s88+awavJ3XT5pF3/joF+vbbb8+4F1XbdLUu8/KxPG2Jl2HERNPiQYiJYroYP3nn39R61S0LdMVh3uei6XSc1l+D8JOet1aU9+RIlfOXv/ylsnPoLt2XQgP5qrzXQVb6TZ0v11UUquhBg9fzyjN/PfdOt2QfdM+QJmtaXR5wwAHmBmyaris8dNWK3vVZFxrUuaNMqnuxFzmUWaa2xMuwYiJL+UdfWQY8lxG3TG6MRl95l6cLbUhU0+KHfcNMGiuk5c9j3rx5JpeYGzMWrCzvydHq1auDa665pvJeI0sFpTuL1n0/Cpd2QhrMp9u9H3nkkSYBfPrppysbNZ9n/vqbBrPdcMMN4ZT2alJd6oqav/3tb2Yw4ObNm4Nrr73WvOuzptcVz2l1r98su0xtixffMZG1/KOvQfd7KiNLjJZZnrbFRB5Nip8q5Jl/l+u1Dl6TI10pMX/+fNMVqODUDa+qpgZLl0JqFLyCwQd1Zypz1YYnulJPNzVTNl+FrPPXhqKrP3S5o8q47bpYl3nVuTxtjBffMdG0eBBioriuxU/W+Xe9XuvgLTlSt64uIbUPCzz22GNNolQHVb4SL10KqaCoW7QbU3dx1YN5dRRfhSzz13rqnhDqNvbdZeo+RkIJcJXaVJd6vIBiXOVR1WmVsrGVtEzDjJeyfMZE0+JB6lqmNsdEHm2JnyyyzD+tXtVJofsOqu1WXGCrOT31uQIAAMAYytVqAAAATUVyBAAA4CA5AgAAcJAcAQAAOEiOAAAAHCRHAAAADpIjAAAAB8kRAACAg+QIAADAQXIEAADgIDkCAABwkBwBAAA4SI4AAAAcJEcAAAAOkiMAAAAHyREAAICD5AgAAMBBcgQAAOAgOQIAAHCQHAEAADhIjgAAABwkRwAAAA6SIwAAAAfJEQAAgIPkCAAAYFoQ/D+LflScRem6ZwAAAABJRU5ErkJggg==\" height=\"170\" width=\"583\"\u003e\u003c/p\u003e\n\u003cp\u003ewhere \u003cem\u003eE\u003c/em\u003e(Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e), \u003cem\u003eE\u003c/em\u003e(Ni) and \u003cem\u003eE\u003c/em\u003e(S)\u0026nbsp;represent the total energy of the NinSn cluster, and the energies of a single Ni and S atom, respectively.\u003c/p\u003e\n\u003cp\u003eIn Figure 5, the \u003cem\u003eE\u003c/em\u003e\u003csub\u003eb\u003c/sub\u003e of Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e\u003cem\u003e\u003csub\u003e\u0026nbsp;\u003c/sub\u003e\u003c/em\u003ecluster\u0026nbsp;for the ground state\u0026nbsp;structures\u0026nbsp;increases monotonically with the size, from\u0026nbsp;2.27\u0026nbsp;eV in\u0026nbsp;Ni\u003csub\u003e1\u003c/sub\u003eS\u003csub\u003e1\u003c/sub\u003e to 4.05 eV in Ni\u003csub\u003e1\u003c/sub\u003e\u003csub\u003e0\u003c/sub\u003eS\u003csub\u003e1\u003c/sub\u003e\u003csub\u003e0\u003c/sub\u003e. This indicates an energetically favorable process in the cluster growth. In comparison, we also calculated the cohesive energy of NiS bulk phase. The calculated cohesive energies are 5.26 eV and 5.24 eV respectively for \u0026beta;-NiS and \u0026alpha;-NiS in a unit cell under low-temperature conditions, slightly larger than other DFT values [52]. Although the binding energy per atom for \u003cem\u003en\u003c/em\u003e\u003cem\u003e\u0026nbsp;\u003c/em\u003e=10 cluster is significantly smaller than the cohesive energy of bulk \u0026beta;-NiS, one can speculate that the \u003cem\u003eE\u003c/em\u003e\u003csub\u003eb\u003c/sub\u003e of Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e\u003cem\u003e\u003csub\u003e\u0026nbsp;\u003c/sub\u003e\u003c/em\u003ecluster will gradually approach the value of solid as the cluster grows toward the bulk. It is noteworthy that the \u003cem\u003eE\u003csub\u003eb\u003c/sub\u003e\u003c/em\u003e of the binary transition metal sulfide Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters is significantly higher than that of pure S\u003csub\u003e2\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e and Ni\u003csub\u003e2\u003cem\u003en\u0026nbsp;\u003c/em\u003e\u003c/sub\u003eclusters at the same cluster sizes[81-83]. For example, the calculated \u003cem\u003eE\u003csub\u003eb\u003c/sub\u003e\u0026nbsp;\u003c/em\u003efor\u0026nbsp;Ni\u003csub\u003e2\u003c/sub\u003eS\u003csub\u003e2\u003c/sub\u003e is 3.28 eV, which is 1.41 eV and 0.98 eV higher than those of Ni\u003csub\u003e4\u003c/sub\u003e (1.87 eV)[82] and S\u003csub\u003e4\u003c/sub\u003e clusters (2.30 eV)[83]. Ni-S interactions are energetically favorable leading to the high stabilization of the binary clusters, suggesting the Ni with partially filled d shells have strong interactions with S atom with unsaturated p electrons.\u003c/p\u003e\n\u003cp\u003eIn addition to the average binding energy, we have calculated the second-order energy difference (\u0026Delta;₂\u003cem\u003eE\u003c/em\u003e) to further evaluate the relative stability of Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters. Figure 5 shows the variation of the \u0026Delta;\u003csub\u003e2\u003c/sub\u003e\u003cem\u003eE\u003c/em\u003e with respect to \u003cem\u003en\u003c/em\u003e for the clusters. From the figure, it is evident that for the Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters, the \u0026Delta;\u003csub\u003e2\u003c/sub\u003e\u003cem\u003eE\u003c/em\u003e exhibits a clear odd-even oscillation pattern, with peaks occurring at \u003cem\u003en\u003c/em\u003e=2, 5, 7, and 9. This suggests that the corresponding clusters have relatively high stability (compared to their neighbors), which is consistent with the conclusions drawn from the binding energy analysis earlier.\u003c/p\u003e\n\u003cp\u003eThe thermodynamic stabilities of the Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters for the ground-state structure were also examined using molecular dynamics simulation (MD) at fixed volume with a thermostat to maintain a constant temperature ensemble (NVT) implemented in the CASTEP code at room temperature (T=300 K). The Nos\u0026eacute;-Hoover-Langevin (NHL) thermostat was used for NVT ensemble[89]. Figure 6 plots the MD results for the Ni\u003csub\u003e5\u003c/sub\u003eS\u003csub\u003e5\u003c/sub\u003e and Ni\u003csub\u003e6\u003c/sub\u003eS\u003csub\u003e6\u003c/sub\u003e clusters. Within 20 \u003cem\u003eps\u003c/em\u003e, changes were\u0026nbsp;undetected in any structures while the instantaneous values of the relative potential energy fluctuate due to thermal fluctuations, but the average values stay constant.\u0026nbsp;Thus, the ground-state clusters of these Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters are real and stable at room temperature.\u003c/p\u003e\n\u003ch2\u003e3.5. Electronic and Magnetic Properties\u003c/h2\u003e\n\u003cp\u003eThe electronic structures of Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters are studied. The deformation electron density (DED) of the Ni\u003csub\u003e6\u003c/sub\u003eS\u003csub\u003e6\u003c/sub\u003e cluster is plotted in Figure 7(a) as an example. Electron density is accumulated around midway of the Ni-S contacts, showing covalent characteristics and robust Ni-S interactions. The interaction between the S-S atoms is negligible. Figure 7(b) depicts the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) pictures of the starting NiS dimer to show electronic structures. In the figure, the electron density of the HOMO and the LUMO states are not only mainly distributed around each Ni and S atoms, but also obviously in the middle of Ni and S atoms. Molecular orbital analysis indicates strong hybridization between the \u003cem\u003ed\u003c/em\u003e orbitals of Ni and the \u003cem\u003ep\u003c/em\u003e orbitals of S. This phenomenon is also reflected in the electronic partial density of states (PDOS) shown in Figure 7(c). The calculated PDOS listed in Figure 7(c) shows that there is an obvious superposition of sharp peaks originating from the d orbital of Ni and p orbital of S from -6 eV to 6 eV energy region near the Fermi level.\u003c/p\u003e\n\u003cp\u003eThe average atomic charge (e) on S atom and HUMO-LUMO gap or bandgap are studied for Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e\u003csub\u003e\u0026nbsp;\u003c/sub\u003elowest-energy\u003csub\u003e\u0026nbsp;\u003c/sub\u003eclusters (\u003cem\u003en\u003c/em\u003e=1-10) and \u0026beta;-NiS bulk phase. Figure 8a shows that charge transfer occurs from Ni to S for each cluster due to a larger electronegativity of the S element. As a result, polarity develops in Ni-S interactions. The mixed ionic-like and covalent bonding are responsible for the high stability of these Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters. In the \u0026beta;-NiS crystalline phase, the charge transfer was 0.093 e/atom, which reveals the charge transfer of clusters is trending toward the bulk parameter with increasing cluster size. A similar trend is also found in the energy gap as shown in Figure 8b. The overall gap energy decreases with \u003cem\u003en\u003c/em\u003e, reaches 0.236 eV at \u003cem\u003en\u003c/em\u003e=10, and eventually vanishes in the crystalline NiS. It is worth noting that there is a peak at \u003cem\u003en\u003c/em\u003e=4 for the gap in Figure\u0026nbsp;8b. This is due to the\u0026nbsp;structural transition\u0026nbsp;from\u0026nbsp;the\u0026nbsp;2D\u0026nbsp;to\u0026nbsp;3D of the Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters. Consequently, the larger clusters approximate closer physicochemical properties concerning the bulk.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTable\u0026nbsp;\u003c/strong\u003e\u003cstrong\u003e2\u003c/strong\u003e The calculated vIP, vEA and chemical hardness (ƞ) of the Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e system.\u003c/p\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n\u003ctable border=\"0\" cellspacing=\"0\" cellpadding=\"0\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003eSystems\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003evIP (eV)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 79px;\"\u003e\n \u003cp\u003evEA (eV)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 75px;\"\u003e\n \u003cp\u003eƞ\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003eNi\u003csub\u003e1\u003c/sub\u003eS\u003csub\u003e1\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e8.79\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 79px;\"\u003e\n \u003cp\u003e0.61\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 75px;\"\u003e\n \u003cp\u003e4.09\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003eNi\u003csub\u003e2\u003c/sub\u003eS\u003csub\u003e2\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e8.14\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 79px;\"\u003e\n \u003cp\u003e1.46\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 75px;\"\u003e\n \u003cp\u003e3.34\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003eNi\u003csub\u003e3\u003c/sub\u003eS\u003csub\u003e3\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e8.37\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 79px;\"\u003e\n \u003cp\u003e2.93\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 75px;\"\u003e\n \u003cp\u003e2.72\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003eNi\u003csub\u003e4\u003c/sub\u003eS\u003csub\u003e4\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e7.19\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 79px;\"\u003e\n \u003cp\u003e1.54\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 75px;\"\u003e\n \u003cp\u003e2.82\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003eNi\u003csub\u003e5\u003c/sub\u003eS\u003csub\u003e5\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e7.16\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 79px;\"\u003e\n \u003cp\u003e1.89\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 75px;\"\u003e\n \u003cp\u003e2.64\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003eNi\u003csub\u003e6\u003c/sub\u003eS\u003csub\u003e6\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e7.22\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 79px;\"\u003e\n \u003cp\u003e2.36\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 75px;\"\u003e\n \u003cp\u003e2.43\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003eNi\u003csub\u003e7\u003c/sub\u003eS\u003csub\u003e7\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e7.31\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 79px;\"\u003e\n \u003cp\u003e2.85\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 75px;\"\u003e\n \u003cp\u003e2.23\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003eNi\u003csub\u003e8\u003c/sub\u003eS\u003csub\u003e8\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e7.25\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 79px;\"\u003e\n \u003cp\u003e3.29\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 75px;\"\u003e\n \u003cp\u003e1.98\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003eNi\u003csub\u003e9\u003c/sub\u003eS\u003csub\u003e9\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e7.17\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 79px;\"\u003e\n \u003cp\u003e3.24\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 75px;\"\u003e\n \u003cp\u003e1.97\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003eNi\u003csub\u003e10\u003c/sub\u003eS\u003csub\u003e10\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e6.90\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 79px;\"\u003e\n \u003cp\u003e3.41\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 75px;\"\u003e\n \u003cp\u003e1.75\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n\u003cp\u003eTo further analyze the electronic stability of the clusters, we have calculated the vertical ionization potential (vIP) and vertical electron affinity (vEA) parameters. The equations and definitions can be found in references [90,91]. The variations of the vIP and vEA parameters as functions of cluster size are shown in Table 2. It is observed that the vIP generally decreases as the cluster size increases, while the vEA first increases for cluster sizes \u003cem\u003en\u003c/em\u003e = 1\u0026ndash;3, reaching a peak of 2.93 at Ni\u003csub\u003e3\u003c/sub\u003eS\u003csub\u003e3\u003c/sub\u003e. From Ni\u003csub\u003e4\u003c/sub\u003eS\u003csub\u003e4\u003c/sub\u003e onward, the vEA again increases gradually. To analyze the chemical stability of the clusters, we determined the chemical hardness (ƞ), which can be derived from the vIP and vEA values. It is found that ƞ decreases monotonically with increasing cluster size, suggesting that larger clusters evolve toward a metallic-like electronic structure. These conclusions are consistent with the findings of Rodr\u0026iacute;guez-Kessler et al. in their study of Pt\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eCu\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e binary clusters [92].\u003c/p\u003e\n\u003cp\u003eMagnetic properties of the predicted clusters are different from the naked Ni peers. Although previous experimental and DFT studies showed that\u0026nbsp;the naked Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters (without stabilizing ligands\u003cu\u003e)\u003c/u\u003e hold the robust magnetic properties[81,93,94], our work predicted coating with shell S\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e cluster will reduce or even eliminate the magnetic properties/magnetism of the core Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters. This is indeed in line with the magnetic properties of the Ni-S systems at a squeezed space[56]. Table 3 shows that the magnetic moment of Ni atoms is completely quenched in the Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e system, except for the dimer NiS and planar triangle Ni\u003csub\u003e3\u003c/sub\u003eS\u003csub\u003e3\u003c/sub\u003e clusters holding magnetic moments of 2\u003cem\u003e\u0026micro;\u003c/em\u003e\u003csub\u003eB\u003c/sub\u003e\u003csub\u003e\u0026nbsp;\u003c/sub\u003eand 4\u003cem\u003e\u0026micro;\u003c/em\u003e\u003csub\u003eB\u003c/sub\u003e\u003csub\u003e\u0026nbsp;\u003c/sub\u003erespectively. The general trend converges to\u0026nbsp;the\u0026nbsp;non-magnetic NiS\u0026nbsp;bulk\u0026nbsp;phase.\u0026nbsp;Similar behavior has been observed in the case of Ni\u0026nbsp;sitting in the center of an Au\u003csub\u003e6\u003c/sub\u003e ring, wherein, the magnetic property of Ni impurity-doped Au\u003csub\u003e6\u003c/sub\u003e cluster is completely quenched[95]. The Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters may not be good candidates for magnetics despite their structural uniqueness.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTable\u0026nbsp;\u003c/strong\u003e\u003cstrong\u003e3\u003c/strong\u003e Total and local magnetic moment (in \u003cem\u003e\u0026micro;\u003c/em\u003e\u003csub\u003eB\u003c/sub\u003e)\u0026nbsp;for the\u0026nbsp;Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e and Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e core clusters calculated in this work with the comparison of the bare Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters from the experiment and other DFT results. The hyphen (-) means no reference data. a) Ref.[97], b) Ref.[98]\u003c/p\u003e\n\u003cdiv align=\"\"\u003e\n \u003ctable border=\"0\" cellspacing=\"0\" cellpadding=\"0\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd colspan=\"3\" style=\"width: 227px;\"\u003e\n \u003cp\u003eThis work\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003eExperiment\u003csup\u003ea)\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003eOther DFT\u003csup\u003eb)\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003eCluster size (\u003cem\u003en\u003c/em\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003eNi\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003eCore Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd colspan=\"2\" style=\"width: 151px;\"\u003e\n \u003cp\u003eBare Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e\u003cem\u003en\u003c/em\u003e=1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e1.100\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e\u003cem\u003en\u003c/em\u003e=2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e-\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e\u003cem\u003en\u003c/em\u003e=3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e2.996\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e-\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e\u003cem\u003en\u003c/em\u003e=4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e-\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e\u003cem\u003en\u003c/em\u003e=5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e9\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e\u003cem\u003en\u003c/em\u003e=6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e9\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e8\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e\u003cem\u003en\u003c/em\u003e=7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e11\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e8\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e\u003cem\u003en\u003c/em\u003e=8\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e13\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e8\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e\u003cem\u003en\u003c/em\u003e=9\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e12\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e8\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e\u003cem\u003en\u003c/em\u003e=10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e13\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e8\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003ch2\u003e3.6. Potential Applications in Hydrogen Storage\u003c/h2\u003e\n\u003cp\u003eThe sulfur atom in the trigonal pyramidal Ni₃S motif exhibits a tendency toward sp\u0026sup3;d\u0026sup2; hybridization in the predicted 3D Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters. This hybridization leaves the S atom with three essentially non-bonding orbitals, manifesting as dangling bonds in the isolated Ni₃S unit. These sites serve as potential active centers for hydrogen anchoring. Figure 4(c) shows the optimized structure of a Ni₃S cluster adsorbing three hydrogen atoms, along with its corresponding molecular orbital (MO) diagram. The results indicate that each hydrogen atom donates one electron to form a stable bond with the central S atom, which in turn bonds covalently with three Ni atoms and three H atoms simultaneously.\u003c/p\u003e\n\u003cp\u003eThe hydrogen storage capacity of Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters is significantly enhanced compared to their individual Ni₃S building blocks, owing to the cumulative effect of multiple such units. To explore this phenomenon further, we selected the Ni₄S₄ cluster as a representative system for systematic hydrogen adsorption analysis. Figure 9 presents the optimized geometries of Ni₄S₄ clusters adsorbing from 1 to 16 hydrogen atoms. Detailed Cartesian coordinates of these configurations are provided in Table S3 in the Supporting Information. During the simulations, hydrogen atoms were incrementally introduced and allowed to fully relax along with the cluster structure until saturation was reached.\u003c/p\u003e\n\u003cp\u003eOur results reveal that chemisorbed hydrogen atoms can bind not only to Ni atoms but also to S atoms. However, Ni atoms exhibit a significantly stronger affinity for hydrogen adsorption than S atoms, as evidenced by the comparison of adsorption energies. For instance, when a single hydrogen atom is adsorbed, the configuration with H bound to a Ni atom is notably more stable than that with H bound to a S atom. Furthermore, when hydrogen atoms are adsorbed on Ni sites, the Ni₄S₄ cluster retains its characteristic square antiprism geometry with only minimal structural deformation. In contrast, adsorption on S sites leads to more substantial distortions. As more hydrogen atoms are adsorbed, the Ni\u0026ndash;S bond lengths increase progressively, and the square antiprism framework becomes increasingly unstable. Beyond 12 hydrogen atoms, the structure approaches the point of collapse, indicating a sharp decline in structural integrity. According to our calculations, Ni₄S₄ can stably adsorb up to 16 hydrogen atoms. Attempts to adsorb more than this result in structural convergence failure or complete disintegration into a disordered configuration.\u003c/p\u003e\n\u003cp\u003eThe gravimetric hydrogen storage capacity of the Ni₄S₄ cluster was calculated using the following formula:\u003c/p\u003e\n\u003cp\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"586\" height=\"85\"\u003e\u003c/p\u003e\n\u003cp\u003ewhere \u003cem\u003en\u003c/em\u003e is the number of hydrogen atoms, \u003cem\u003eM\u003c/em\u003e\u003csub\u003eH\u003c/sub\u003e is the atomic mass of hydrogen, and \u003cem\u003eM\u003c/em\u003e\u003csub\u003ecluster\u003c/sub\u003e is the total mass of the pristine cluster.\u003c/p\u003e\n\u003cp\u003eAs summarized in Table 4, the hydrogen storage capacity increases from 0.3 wt% for a single H atom to a maximum of 4.3 wt% for 16 H atoms. Although this value does not yet meet the U.S. Department of Energy (DOE) 2025 target of 5.5 wt% for onboard hydrogen storage systems, it shows promising potential. Notably, recent work by Rodr\u0026iacute;guez-Kessler et al. has highlighted the viability of binary clusters as efficient hydrogen storage media, which supports the conclusions drawn in our study regarding the application potential of Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters in clean energy technologies [96\u0026ndash;99].\u003c/p\u003e\n\u003cp\u003eTo further understand the nature of hydrogen-cluster interactions, we analyzed the average bond lengths between hydrogen and Ni/S atoms, along with Mulliken charge distributions on the adsorbed hydrogen atoms. As shown in Table 4, the average H\u0026ndash;S bond length increases from 1.39 \u0026Aring; to 1.47 \u0026Aring;, while the H\u0026ndash;Ni bond length grows from approximately 1.47 \u0026Aring; to 1.60 \u0026Aring; as the number of hydrogen atoms increases. This bond elongation suggests weakened chemisorption strength at higher hydrogen coverage. Mulliken charge analysis also shows a decrease in charge transfer from the cluster to hydrogen atoms with increased loading, corroborating the observed trend in reduced interaction strength.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTable\u0026nbsp;\u003c/strong\u003e\u003cstrong\u003e4\u003c/strong\u003e Calculated hydrogen weight percentage (wt%), average bond distances (\u0026Aring;) of H\u0026ndash;Ni and H\u0026ndash;S, and average Mulliken charge (a.u.) on the hydrogen atom for the Ni₄S₄ cluster adsorbing 1 to 16 hydrogen atoms. A hyphen (\u0026ndash;) indicates unavailable data.\u003c/p\u003e\n\u003cdiv align=\"\"\u003e\n \u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"455\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd rowspan=\"2\" valign=\"top\" style=\"width: 85px;\"\u003e\n \u003cp\u003eNumber of the H atoms\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 99px;\"\u003e\n \u003cp\u003eHydrogen weight percent\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd colspan=\"3\" valign=\"top\" style=\"width: 126px;\"\u003e\n \u003cp\u003eBond length\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 23px;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd colspan=\"2\" valign=\"top\" style=\"width: 122px;\"\u003e\n \u003cp\u003eCharge\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 99px;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd colspan=\"2\" valign=\"top\" style=\"width: 61px;\"\u003e\n \u003cp\u003eH-Ni\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 65px;\"\u003e\n \u003cp\u003eH-S\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 23px;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 56px;\"\u003e\n \u003cp\u003eH-Ni\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 66px;\"\u003e\n \u003cp\u003eH-S\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 99px;\"\u003e\n \u003cp\u003e0.3%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 54px;\"\u003e\n \u003cp\u003e1.471\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd colspan=\"2\" style=\"width: 72px;\"\u003e\n \u003cp\u003e-\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 23px;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 56px;\"\u003e\n \u003cp\u003e0.088\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003e-\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 99px;\"\u003e\n \u003cp\u003e0.6%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 54px;\"\u003e\n \u003cp\u003e1.472\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd colspan=\"2\" style=\"width: 72px;\"\u003e\n \u003cp\u003e-\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 23px;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 56px;\"\u003e\n \u003cp\u003e0.085\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003e-\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 99px;\"\u003e\n \u003cp\u003e0.8%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 54px;\"\u003e\n \u003cp\u003e1.472\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd colspan=\"2\" style=\"width: 72px;\"\u003e\n \u003cp\u003e1.393\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 23px;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 56px;\"\u003e\n \u003cp\u003e0.080\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003e0.233\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 99px;\"\u003e\n \u003cp\u003e1.1%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 54px;\"\u003e\n \u003cp\u003e1.473\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd colspan=\"2\" style=\"width: 72px;\"\u003e\n \u003cp\u003e1.394\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 23px;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 56px;\"\u003e\n \u003cp\u003e0.075\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003e0.221\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 99px;\"\u003e\n \u003cp\u003e1.4%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 54px;\"\u003e\n \u003cp\u003e1.476\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd colspan=\"2\" style=\"width: 72px;\"\u003e\n \u003cp\u003e1.396\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 23px;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 56px;\"\u003e\n \u003cp\u003e0.073\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003e0.218\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 99px;\"\u003e\n \u003cp\u003e1.7%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 54px;\"\u003e\n \u003cp\u003e1.478\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd colspan=\"2\" style=\"width: 72px;\"\u003e\n \u003cp\u003e1.397\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 23px;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 56px;\"\u003e\n \u003cp\u003e0.069\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003e0.208\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 99px;\"\u003e\n \u003cp\u003e1.9%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 54px;\"\u003e\n \u003cp\u003e1.479\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd colspan=\"2\" style=\"width: 72px;\"\u003e\n \u003cp\u003e1.398\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 23px;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 56px;\"\u003e\n \u003cp\u003e0.070\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003e0.210\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e8\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 99px;\"\u003e\n \u003cp\u003e2.2%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 54px;\"\u003e\n \u003cp\u003e1.483\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd colspan=\"2\" style=\"width: 72px;\"\u003e\n \u003cp\u003e1.401\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 23px;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 56px;\"\u003e\n \u003cp\u003e0.062\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003e0.206\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e9\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 99px;\"\u003e\n \u003cp\u003e2.5%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 54px;\"\u003e\n \u003cp\u003e1.486\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd colspan=\"2\" style=\"width: 72px;\"\u003e\n \u003cp\u003e1.409\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 23px;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 56px;\"\u003e\n \u003cp\u003e0.058\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003e0.201\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 99px;\"\u003e\n \u003cp\u003e2.8%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 54px;\"\u003e\n \u003cp\u003e1.492\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd colspan=\"2\" style=\"width: 72px;\"\u003e\n \u003cp\u003e1.412\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 23px;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 56px;\"\u003e\n \u003cp\u003e0.059\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003e0.186\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e11\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 99px;\"\u003e\n \u003cp\u003e3.0%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 54px;\"\u003e\n \u003cp\u003e1.510\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd colspan=\"2\" style=\"width: 72px;\"\u003e\n \u003cp\u003e1.425\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 23px;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 56px;\"\u003e\n \u003cp\u003e0.055\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003e0.171\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e12\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 99px;\"\u003e\n \u003cp\u003e3.3%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 54px;\"\u003e\n \u003cp\u003e1.530\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd colspan=\"2\" style=\"width: 72px;\"\u003e\n \u003cp\u003e1.432\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 23px;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 56px;\"\u003e\n \u003cp\u003e0.048\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003e0.162\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e13\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 99px;\"\u003e\n \u003cp\u003e3.6%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 54px;\"\u003e\n \u003cp\u003e1.552\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd colspan=\"2\" style=\"width: 72px;\"\u003e\n \u003cp\u003e1.447\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 23px;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 56px;\"\u003e\n \u003cp\u003e0.045\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003e0.153\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e14\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 99px;\"\u003e\n \u003cp\u003e3.8%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 54px;\"\u003e\n \u003cp\u003e1.575\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd colspan=\"2\" style=\"width: 72px;\"\u003e\n \u003cp\u003e1.452\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 23px;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 56px;\"\u003e\n \u003cp\u003e0.042\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003e0.146\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e15\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 99px;\"\u003e\n \u003cp\u003e4.1%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 54px;\"\u003e\n \u003cp\u003e1.583\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd colspan=\"2\" style=\"width: 72px;\"\u003e\n \u003cp\u003e1.469\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 23px;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 56px;\"\u003e\n \u003cp\u003e0.037\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003e0.138\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e16\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 99px;\"\u003e\n \u003cp\u003e4.3%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 54px;\"\u003e\n \u003cp\u003e1.596\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd colspan=\"2\" style=\"width: 72px;\"\u003e\n \u003cp\u003e1.473\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 23px;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 56px;\"\u003e\n \u003cp\u003e0.035\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003e0.129\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e"},{"header":"4. CONCLUSIONS AND SUMMARY","content":"\u003cp\u003eIn conclusion, a successful disassembly-assembly strategy has been revealed in the structural optimizations of the binary Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e (\u003cem\u003en\u003c/em\u003e=1-10) clusters and paved the way to study cluster properties. Based on the DFT results, the assembling paths of Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e systems are simplified in inlaying the core Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e\u003csub\u003e\u0026nbsp;\u003c/sub\u003eto shell S\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters within the same geometric point group. Following this discovery, investigations of the structural evolution of the compound clusters are reduced to probations of component Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e or S\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e systems with simpler structures. The NiS cluster can follow the core-shell constructions to reach the crystalline phase. The specific orbital hybridization from the bonding S atoms is proposed in charge of the spatial extension of the clusters, and the assembly-disassembly scheme for the TMD clusters. Despite magnetic mediocrity, the thermostable Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters are endowed with strong abilities to absorb hydrogen atoms for clean energy storage. Starting from the explicated NiS case, the present work is hoped to benefit the overall cluster sciences by serving a fast and efficient shortcut to identifying the most probable isomers of large clusters complicated in structure and rich in element species.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eWeide Liu, Fenfa Zhang, Bowei Jin, Xuezhou Lu, and Zhengyang Ding wrote the main manuscript text. Xiao Wang and Meng Zhang contributed to the conceptualization and manuscript revision. All authors reviewed and approved the final manuscript.\u003c/p\u003e\u003cp\u003e\u003cstrong\u003eSupporting Information Available:\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe benchmark calculation on NiS dimer using a different type of functionals. Details of structures and x,y,z coordinates of the lowest-energy structures and low-lying isomers of Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e (\u003cem\u003en\u003c/em\u003e=1-10) clusters, as well as the Ni\u003csub\u003e4\u003c/sub\u003eS\u003csub\u003e4\u003c/sub\u003e cluster with adsorbed hydrogen atoms. The shortest and average Ni-S bond lengths of the Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e clusters (\u003cem\u003en\u003c/em\u003e=1-10) and \u0026beta;-NiS bulk phase. These materials are available free of charge via the Internet.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eNotes\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors declare no competing financial interest.\u0026nbsp;\u003cbr\u003e\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eACKNOWLEDGMENTS\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis work is financially supported by the University Student Innovation Program of China (Grant No. 201810251078).\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003eRohmer M. M.; B\u0026eacute;nard M.; Poblet J. M. Structure, reactivity, and growth pathways of metallocarbohedrenes M8C12 and transition metal/carbon clusters and nanocrystals: A challenge to computational chemistry. \u003cem\u003eChem. Rev.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e2000\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;100\u003c/em\u003e, 495-542.\u003c/li\u003e\n \u003cli\u003eEinax M.; Dieterich W.; Maass P. Colloquium: Cluster growth on surfaces: Densities, size distributions, and morphologies.\u003cem\u003e\u0026nbsp;Rev. Mod. Phys.\u003c/em\u003e\u003cstrong\u003e\u0026nbsp;2013\u003c/strong\u003e,\u003cstrong\u003e\u0026nbsp;\u003c/strong\u003e\u003cem\u003e85\u003c/em\u003e, 921.\u003c/li\u003e\n \u003cli\u003ePauna H.; Shi X.; Huttula M.; et al. Evolution of lithium clusters to superatomic Li\u003cem\u003e\u003csub\u003e3\u003c/sub\u003e\u003c/em\u003eO+. \u003cem\u003eAppl. Phys. Lett.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e2017\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;111\u003c/em\u003e, 103901.\u003c/li\u003e\n \u003cli\u003eZubarev D. Y.; Averkiev B. B.; Zhai H. J. et al. Aromaticity and antiaromaticity in transition-metal systems. \u003cem\u003ePhys. Chem. Chem. Phys.\u003c/em\u003e\u003cstrong\u003e\u0026nbsp;2008\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;10\u003c/em\u003e, 257-267.\u003c/li\u003e\n \u003cli\u003eKroto H. W.; Heath J. R.; O\u0026apos;Brien S. C.; et al. C60: Buckminsterfullerene. \u003cem\u003eNature\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e1985\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;318\u003c/em\u003e, 162-163.\u003c/li\u003e\n \u003cli\u003eReveles J. U.; Clayborne P. A.; Reber A. C.; et al. Designer magnetic superatoms. \u003cem\u003eNat. Chem.,\u003c/em\u003e\u003cstrong\u003e\u0026nbsp;2009\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;1\u003c/em\u003e, 310.\u003c/li\u003e\n \u003cli\u003eWang X.; Adeleke A. A.; Cao W.; et al. Structures of Nanoalloy Clusters Au\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e Al\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e (n=1\u0026ndash;10) and the Growth Patterns to the Bulk Phase.\u003cem\u003e\u0026nbsp;J. Phys. Chem. C,\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e2016\u003c/strong\u003e, \u003cem\u003e120\u003c/em\u003e, 25588-25595.\u003c/li\u003e\n \u003cli\u003eGong X. G. Structure and stability of cluster-assembled solid Al12C(Si): a first-principles study. \u003cem\u003ePhys. Rev. B,\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e1997\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;56\u003c/em\u003e, 1091.\u003c/li\u003e\n \u003cli\u003eWang L. M.; Wang L. S. Probing the electronic properties and structural evolution of anionic gold clusters in the gas phase. \u003cem\u003eNanoscale\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e2012\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;4\u003c/em\u003e, 4038-4053.\u003c/li\u003e\n \u003cli\u003eHeine T. Transition metal chalcogenides: ultrathin inorganic materials with tunable electronic properties. \u003cem\u003eAccounts. Chem. Res.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e2015\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;48\u003c/em\u003e, 65-72.\u003c/li\u003e\n \u003cli\u003eSoldatov A. V.; Kravtsova A. N.; Fleet M. E.; et al. Electronic structure of MeS (Me= Ni, Co, Fe): x-ray absorption analysis. \u003cem\u003eJ. Phys-Condens. Mat.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e2004\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;16\u003c/em\u003e, 7545.\u003c/li\u003e\n \u003cli\u003eChianelli R. R.; Daage M.; Ledoux M. J. Fundamental studies of transition-metal sulfide catalytic materials, Advances in Catalysis.\u003cem\u003e\u0026nbsp;Academic Press,\u003c/em\u003e\u003cstrong\u003e\u0026nbsp;1994\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;40\u003c/em\u003e, 177-232.\u003c/li\u003e\n \u003cli\u003eRohrbach A.; Hafner J.; Kresse G. Electronic correlation effects in transition-metal sulfides. \u003cem\u003eJ. Phys-Condens. Mat.\u003c/em\u003e\u003cstrong\u003e\u0026nbsp;2003\u003c/strong\u003e, \u003cem\u003e15\u003c/em\u003e, 979.\u003c/li\u003e\n \u003cli\u003eChianelli R. R.; Berhault G.; Torres B. Unsupported transition metal sulfide catalysts: 100 years of science and application. \u003cem\u003eCatal. Today.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e2009\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;147\u003c/em\u003e, 275-286.\u003c/li\u003e\n \u003cli\u003eEl Nakat J. H.; Dance I. G.; Fisher K. J.; et al. Laser-ablation FTICR (Fourier-transform ICR) mass spectrometry of metal sulfides: gaseous anionic nickel-sulfur [Ni\u003cem\u003e\u003csub\u003ex\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003ey\u003c/sub\u003e\u003c/em\u003e] clusters. \u003cem\u003eJ. Am. Chem. Soc.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e1991\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;113\u003c/em\u003e, 5141-5148.\u003c/li\u003e\n \u003cli\u003eEl Nakat J. H.; Dance I. G.; Fisher K. J.; et al. Gas phase copper chalcogenide cluster ions, formed by laser-ablation. \u003cem\u003eInorg. Chem.\u003cstrong\u003e\u0026nbsp;\u003c/strong\u003e\u003c/em\u003e\u003cstrong\u003e1991\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;30\u003c/em\u003e, 2957-2958.\u003c/li\u003e\n \u003cli\u003eEl Nakat J.; Dance I.; Fisher K.; et al. Gas-phase silver chalcogenide ions investigated by laser-ablation Fourier transform ion cyclotron resonance mass spectrometry. \u003cem\u003eJ. Chem. Soc., Chem. Commun.\u003c/em\u003e\u003cstrong\u003e\u0026nbsp;1991\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;11\u003c/em\u003e, 746-748.\u003c/li\u003e\n \u003cli\u003eEl Nakat J. H.; Fisher K. J.; Dance I. G.; et al. Gas phase metal chalcogenide cluster ions: a new cobalt-sulfur [CoxSy]-series up to [Co\u003cem\u003e\u003csub\u003e38\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003e24\u003c/sub\u003e\u003c/em\u003e]-and two iron-sulfur [Fe\u003cem\u003e\u003csub\u003ex\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003ey\u003c/sub\u003e\u003c/em\u003e]-series. \u003cem\u003eInorg. Chem.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e1993\u003c/strong\u003e, \u003cem\u003e32\u003c/em\u003e, 1931-1940.\u003c/li\u003e\n \u003cli\u003eDance I. G.; Fisher K. J.; Willett G. D. Reactions of 29 transition metal cations, in the same oxidation state and under the same gas-phase conditions, with sulfur. Inorg. \u003cem\u003eChem.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e1996\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;35\u003c/em\u003e, 4177-4184.\u003c/li\u003e\n \u003cli\u003eFisher K.; Dance I.; Willett G.; et al. Gas-phase metal sulfide cluster anions. \u003cem\u003eJ. Chem. Soc., Dalton Trans.\u003c/em\u003e\u003cstrong\u003e\u0026nbsp;1996\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;5\u003c/em\u003e, 709-718.\u003c/li\u003e\n \u003cli\u003eDance I. G,; Fisher K. J.; Willett G. D. Molecular manganese sulfide clusters formed by laser ablation.\u003cem\u003e\u0026nbsp;J. Chem. Soc., Dalton Trans.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e1997\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;15\u003c/em\u003e, 2557-2562.\u003c/li\u003e\n \u003cli\u003eZhang N.; Yu Z.; Wu X.; et al. Formation and photolysis of tantalum sulfide cluster ions. \u003cem\u003eJ. Chem. Soc., Faraday Trans.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e1993\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;89\u003c/em\u003e, 1779-1782.\u003c/li\u003e\n \u003cli\u003eYu Z.; Zhang N.; Wu X.; et al. The production and photodissociation of iron\u0026ndash;sulfur cluster ions.\u003cem\u003e\u0026nbsp;J. Chem. Phys.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e1993\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;99\u003c/em\u003e, 1765-1770.\u003c/li\u003e\n \u003cli\u003eShi Y.; Zhang N.; Gao Z.; et al. The formation, photodissociation, and bond structure of cobalt\u0026ndash;sulfur cluster ions. \u003cem\u003eJ. Chem. Phys.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e1994\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;101\u003c/em\u003e, 9528-9533.\u003c/li\u003e\n \u003cli\u003eLiu Y.; Xing J.; Fu H.; et al. Structural stability, mechanical properties, electronic structures and thermal properties of XS (X= Ti, V, Cr, Mn, Fe, Co, Ni) binary compounds. \u003cem\u003ePhys. Lett. A\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e2017\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;381\u003c/em\u003e, 2648-2657.\u003c/li\u003e\n \u003cli\u003eLiang B.; Andrews L. Infrared spectra and density functional theory calculations of group 4 transition metal sulfides. \u003cem\u003eJ. Phys. Chem. A\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e2002\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;106\u003c/em\u003e, 6295-6301.\u003c/li\u003e\n \u003cli\u003eLiang B.; Andrews L. Matrix Infrared Spectra and Quasirelativistic DFT Studies of ThS and ThS\u003cem\u003e\u003csub\u003e2\u003c/sub\u003e\u003c/em\u003e. \u003cem\u003eJ. Phys. Chem. A\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e2002\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;106\u003c/em\u003e, 4038-4041.\u003c/li\u003e\n \u003cli\u003eLiang B.; Andrews L. Infrared spectra and density functional theory calculations of group 6 transition metal sulfides in solid argon. \u003cem\u003eJ. Phys. Chem. A\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e2002\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;106\u003c/em\u003e, 6945-6951.\u003c/li\u003e\n \u003cli\u003eLiang B.; Wang X.; Andrews L. Infrared spectra and density functional theory calculations of group 8 transition metal sulfide molecules. \u003cem\u003eJ. Phys. Chem. A\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e2009\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;113\u003c/em\u003e, 5375-5384.\u003c/li\u003e\n \u003cli\u003eLiang B.; Wang X.; Andrews L. Infrared spectra and density functional theory calculations of group 10 transition metal sulfide molecules and complexes. \u003cem\u003eJ. Phys. Chem. A\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e2009\u003c/strong\u003e, \u003cem\u003e113\u003c/em\u003e, 3336-3343.\u003c/li\u003e\n \u003cli\u003eZhao Y. C.; Yuan J.; Zhang Z, G.; et al. Structures of manganese polysulfides: mass-selected photodissociation and density functional calculations. \u003cem\u003eDalton Trans.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e2011\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;40\u003c/em\u003e, 2502-2508.\u003c/li\u003e\n \u003cli\u003eKenzler S.; Schrenk C.; Frojd A. R.; et al. Au\u003cem\u003e\u003csub\u003e70\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003e20\u003c/sub\u003e\u003c/em\u003e(PPh\u003cem\u003e\u003csub\u003e3\u003c/sub\u003e\u003c/em\u003e)\u003cem\u003e\u003csub\u003e12\u003c/sub\u003e\u003c/em\u003e: an intermediate sized metalloid gold cluster stabilized by the Au\u003cem\u003e\u003csub\u003e4\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003e4\u003c/sub\u003e\u003c/em\u003e ring motif and Au-PPh\u003cem\u003e\u003csub\u003e3\u003c/sub\u003e\u003c/em\u003e groups. \u003cem\u003eChem. Commun.\u003cstrong\u003e\u0026nbsp;\u003c/strong\u003e\u003c/em\u003e\u003cstrong\u003e2018\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;54\u003c/em\u003e, 248-251.\u003c/li\u003e\n \u003cli\u003ePei Y.; Gao Y.; Shao N.; et al. Thiolate-protected Au\u003cem\u003e\u003csub\u003e20\u003c/sub\u003e\u003c/em\u003e(SR)\u003cem\u003e\u003csub\u003e16\u003c/sub\u003e\u003c/em\u003e cluster: prolate Au\u003cem\u003e\u003csub\u003e8\u003c/sub\u003e\u003c/em\u003e core with new [Au\u003cem\u003e\u003csub\u003e3\u003c/sub\u003e\u003c/em\u003e(SR)\u003cem\u003e\u003csub\u003e4\u003c/sub\u003e\u003c/em\u003e] staple motif .\u003cem\u003e\u0026nbsp;J. Am. Chem. Soc.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e2009\u003c/strong\u003e, \u003cem\u003e131\u003c/em\u003e, 13619-13621.\u003c/li\u003e\n \u003cli\u003ePei Y.; Tang J.; Tang X.; et al. New structure model of Au\u003cem\u003e\u003csub\u003e22\u003c/sub\u003e\u003c/em\u003e(SR)\u003cem\u003e\u003csub\u003e18\u003c/sub\u003e\u003c/em\u003e: bitetrahederon golden kernel enclosed by [Au6(SR)6]Au(I) complex.\u003cem\u003e\u0026nbsp;J. Phys. Chem. Lett.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e2015\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;6\u003c/em\u003e, 1390-1395.\u003c/li\u003e\n \u003cli\u003e\u003cstrong\u003eMondal A.; Vomiero A.\u003c/strong\u003e 2D transition metal dichalcogenides-based electrocatalysts for hydrogen evolution reaction. \u003cem\u003eAdv. Funct. Mater.\u003c/em\u003e \u003cstrong\u003e2022\u003c/strong\u003e, \u003cem\u003e32\u003c/em\u003e(52), 2208994.\u003c/li\u003e\n \u003cli\u003e\u003cstrong\u003eZhang J.; Wu J.; Guo H.; et al.\u003c/strong\u003e Unveiling active sites for the hydrogen evolution reaction on monolayer MoS₂. \u003cem\u003eAdv. Mater.\u003c/em\u003e \u003cstrong\u003e2017\u003c/strong\u003e, \u003cem\u003e29\u003c/em\u003e(42), 1701955.\u003c/li\u003e\n \u003cli\u003e\u003cstrong\u003eVoiry D.; Salehi M.; Silva R.; et al.\u003c/strong\u003e Conducting MoS₂ nanosheets as catalysts for hydrogen evolution reaction. \u003cem\u003eNano Lett.\u003c/em\u003e \u003cstrong\u003e2013\u003c/strong\u003e, \u003cem\u003e13\u003c/em\u003e(12), 6222-6227.\u003c/li\u003e\n \u003cli\u003e\u003cstrong\u003eLi G.; Zhang D.; Qiao Q.; et al.\u003c/strong\u003e All the catalytic active sites of MoS₂ for hydrogen evolution. \u003cem\u003eJ. Am. Chem. Soc.\u003c/em\u003e \u003cstrong\u003e2016\u003c/strong\u003e, \u003cem\u003e138\u003c/em\u003e(51), 16632-16638.\u003c/li\u003e\n \u003cli\u003e\u003cstrong\u003eCao Y.\u003c/strong\u003e Roadmap and direction toward high-performance MoS₂ hydrogen evolution catalysts. \u003cem\u003eACS Nano\u003c/em\u003e \u003cstrong\u003e2021\u003c/strong\u003e, \u003cem\u003e15\u003c/em\u003e(7), 11014-11039.\u003c/li\u003e\n \u003cli\u003e\u003cstrong\u003eLi Y.; Wang H.; Xie L.; et al.\u003c/strong\u003e MoS₂ nanoparticles grown on graphene: an advanced catalyst for the hydrogen evolution reaction. \u003cem\u003eJ. Am. Chem. Soc.\u003c/em\u003e \u003cstrong\u003e2011\u003c/strong\u003e, \u003cem\u003e133\u003c/em\u003e(19), 7296-7299.\u003c/li\u003e\n \u003cli\u003e\u003cstrong\u003eCheng Y.; Lu S.; Liao F.; et al.\u003c/strong\u003e Rh-MoS₂ nanocomposite catalysts with Pt-like activity for hydrogen evolution reaction. \u003cem\u003eAdv. Funct. Mater.\u003c/em\u003e \u003cstrong\u003e2017\u003c/strong\u003e, \u003cem\u003e27\u003c/em\u003e(23), 1700359.\u003c/li\u003e\n \u003cli\u003e\u003cstrong\u003eYu E.; Pan Y.\u003c/strong\u003e Enhancing the catalytic hydrogen evolution reaction (HER) of the defective borophene@Pt/Pd/MoS₂ heterojunction. \u003cem\u003eInt. J. Hydrogen Energy\u003c/em\u003e \u003cstrong\u003e2024\u003c/strong\u003e, \u003cem\u003e50\u003c/em\u003e, 920-931.\u003c/li\u003e\n \u003cli\u003e\u003cstrong\u003eDuan H.; Wang C.; Li G.; et al.\u003c/strong\u003e Single-atom-layer catalysis in a MoS₂ monolayer activated by long-range ferromagnetism for the hydrogen evolution reaction: beyond single-atom catalysis. \u003cem\u003eAngew. Chem.\u003c/em\u003e \u003cstrong\u003e2021\u003c/strong\u003e, \u003cem\u003e133\u003c/em\u003e(13), 7327-7334.\u003c/li\u003e\n \u003cli\u003eGibbs G.; V.; Downs R. T.; Prewitt C. T.; et al. Electron Density Distributions Calculated for the Nickel Sulfides Millerite, Vaesite, and Heazlewoodite and Nickel Metal: A Case for the Importance of Ni\u0026minus; Ni Bond Paths for Electron Transport. \u003cem\u003eJ. Phys. Chem. B\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e2005\u003c/strong\u003e, \u003cem\u003e109\u003c/em\u003e, 21788-21795.\u003c/li\u003e\n \u003cli\u003eMetcalf P. A.; Crooker B. C.; McElfresh M.; et al. Low-temperature electronic and magnetic properties of single-crystal Ni\u003cem\u003e\u003csub\u003e3\u003c/sub\u003e\u003c/em\u003eS\u003cem\u003e\u003csub\u003e2\u003c/sub\u003e\u003c/em\u003e. \u003cem\u003ePhys. Rev. B\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e1994\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;50\u003c/em\u003e, 2055.\u003c/li\u003e\n \u003cli\u003eTyler J. M.; Fry J. L. Energy bands of hexagonal NiS. \u003cem\u003ePhys. Rev. B\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e1970\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;1\u003c/em\u003e, 4604.\u003c/li\u003e\n \u003cli\u003eKullerud G.; Yund R. A. The Ni-S system and related minerals. \u003cem\u003eJ. Petrol.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e1962\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;3\u003c/em\u003e, 126-175.\u003c/li\u003e\n \u003cli\u003eSowa H.; Ahsbahs H.; Schmitz W. X-ray diffraction studies of millerite NiS under non-ambient conditions. \u003cem\u003ePhys. Chem. Miner.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e2004\u003c/strong\u003e\u003cem\u003e, 31\u003c/em\u003e, 321-327.\u003c/li\u003e\n \u003cli\u003eGoh S. W.; Buckley A. N.; Lamb R. N.; et al. Sulfur electronic environments in \u0026alpha;-NiS and \u0026beta;-NiS: examination of the relationship between coordination number and core electron binding energies. \u003cem\u003ePhys. Chem. Miner.\u003c/em\u003e \u003cstrong\u003e2006\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;33\u003c/em\u003e, 98-105.\u003c/li\u003e\n \u003cli\u003eKrishnakumar S. R.; Shanthi N.; Sarma D. D. Electronic structure of millerite NiS. \u003cem\u003ePhys. Rev. B\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e2002\u003c/strong\u003e\u003cem\u003e, 66\u003c/em\u003e, 115105.\u003c/li\u003e\n \u003cli\u003eWang H.; Pring A.; Ngothai Y.; et al. The kinetics of the\u0026nbsp;\u0026alpha;\u0026rarr;\u0026nbsp;\u0026beta;\u0026nbsp;transition in synthetic nickel monosulfide. \u003cem\u003eAm. Mineral.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e2006\u003c/strong\u003e\u003cem\u003e, 91\u003c/em\u003e, 171-181.\u003c/li\u003e\n \u003cli\u003eRaybaud P.; Kresse G.; Hafner J.; et al. Ab initio density functional studies of transition-metal sulphides: I. Crystal structure and cohesive properties. \u003cem\u003eJ. Phys-Condens. Mat.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e1997\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;9\u003c/em\u003e, 11085.\u003c/li\u003e\n \u003cli\u003eRaybaud P.; Hafner J.; Kresse G.; et al. Ab initio density functional studies of transition-metal sulphides: II. Electronic structure. \u003cem\u003eJ. Phys-Condens. Mat.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e1997\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;9\u003c/em\u003e, 11107.\u003c/li\u003e\n \u003cli\u003eWang J. H.; Cheng Z.; Br\u0026eacute;das J. L.; et al. Electronic and vibrational properties of nickel sulfides from first principles. \u003cem\u003eJ. Chem. Phys.\u003c/em\u003e \u003cstrong\u003e2007\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;127\u003c/em\u003e, 214705.\u003c/li\u003e\n \u003cli\u003eMusselman I. H.; Linton R. W.; Simons D. S. Cluster ion formation under laser bombardment. Studies of recombination using isotope labeling. \u003cem\u003eAnal. Chem.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e1988\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;60\u003c/em\u003e, 110-114.\u003c/li\u003e\n \u003cli\u003eShi X.; Posysaev S.; Huttula M.; et al. Metallic contact between MoS\u003cem\u003e\u003csub\u003e2\u003c/sub\u003e\u003c/em\u003e and Ni via Au nanoglue.\u003cem\u003e\u0026nbsp;Small\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e2018\u003c/strong\u003e, \u003cem\u003e14\u003c/em\u003e, 1704526.\u003c/li\u003e\n \u003cli\u003eDeng S.; Zhang K.; Xie D.; et al. High-index-faceted Ni3S2 branch arrays as bifunctional electrocatalysts for efficient water splitting. Nano-Micro. \u003cem\u003eLett.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e2019\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;11\u003c/em\u003e, 12.\u003c/li\u003e\n \u003cli\u003ePanda C.; Menezes P. W.; Yao S.; et al. Boosting Electrocatalytic Hydrogen Evolution Activity with a NiPt\u003cem\u003e\u003csub\u003e3\u003c/sub\u003e\u003c/em\u003e@NiS Heteronanostructure Evolved from a Molecular Nickel\u0026ndash;Platinum Precursor.\u003cem\u003e\u0026nbsp;J. Am. Chem. Soc.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e2019\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;141\u003c/em\u003e, 13306-13310.\u003c/li\u003e\n \u003cli\u003eWang Y.; Lv J.; Zhu L.; et al. CALYPSO: A method for crystal structure prediction. \u003cem\u003eComput. Phys. Commun.\u003c/em\u003e \u003cstrong\u003e2012\u003c/strong\u003e, \u003cem\u003e183\u003c/em\u003e, 2063-2070.\u003c/li\u003e\n \u003cli\u003eWang Y.; Miao M.; Lv J.; et al. An effective structure prediction method for layered materials based on 2D particle swarm optimization algorithm. \u003cem\u003eJ. Chem. Phys.\u003cstrong\u003e\u0026nbsp;\u003c/strong\u003e\u003c/em\u003e\u003cstrong\u003e2012\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;137\u003c/em\u003e, 224108.\u003c/li\u003e\n \u003cli\u003eLv J.; Wang Y.; Zhu L.; et al. Particle-swarm structure prediction on clusters. \u003cem\u003eJ. Chem. Phys.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e2012\u003c/strong\u003e\u003cem\u003e, 137\u003c/em\u003e, 084104.\u003c/li\u003e\n \u003cli\u003eFrisch M. J.; Trucks G. W.; Schlegel H. B.; et al. Gaussian 09. vol. Revision D. 01, Gaussian. \u003cem\u003eInc., Wallingford, CT\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e2009\u003c/strong\u003e.\u003c/li\u003e\n \u003cli\u003eStevens W. J.; Basch H.; Krauss M. Compact effective potentials and efficient shared‐exponent basis sets for the first‐and second‐row atoms. \u003cem\u003eJ. Chem. Phys.\u003c/em\u003e\u003cstrong\u003e\u0026nbsp;1984\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;81\u003c/em\u003e, 6026-6033.\u003c/li\u003e\n \u003cli\u003eHay P. J.; Wadt W. R. Ab initio effective core potentials for molecular calculations. Potentials for the transition metal atoms Sc to Hg. \u003cem\u003eJ. Chem. Phys.\u003cstrong\u003e\u0026nbsp;\u003c/strong\u003e\u003c/em\u003e\u003cstrong\u003e1985\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;82\u003c/em\u003e, 270-283.\u003c/li\u003e\n \u003cli\u003eDelley B. An all‐electron numerical method for solving the local density functional for polyatomic molecules. \u003cem\u003eJ. Chem. Phys.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e1990\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;92\u003c/em\u003e, 508-517.\u003c/li\u003e\n \u003cli\u003ePerdew J. P.; Burke K.; Ernzerhof M. Generalized gradient approximation made simple. \u003cem\u003ePhys. Rev. Lett.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e1996\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;77\u003c/em\u003e, 3865.\u003c/li\u003e\n \u003cli\u003eBecke A. D. Density-functional exchange-energy approximation with correct asymptotic behavior. \u003cem\u003ePhys. Rev. Appl.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e1988\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;38\u003c/em\u003e, 3098.\u003c/li\u003e\n \u003cli\u003eLee C.; Yang W.; Parr R. G. Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density.\u003cem\u003e\u0026nbsp;Phys. Rev. B\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e1988\u003c/strong\u003e, \u003cem\u003e37\u003c/em\u003e, 785.\u003c/li\u003e\n \u003cli\u003ePerdew J. P.; Wang Y. Accurate and simple analytic representation of the electron-gas correlation energy. \u003cem\u003ePhys. Rev. B\u003cstrong\u003e\u0026nbsp;\u003c/strong\u003e\u003c/em\u003e\u003cstrong\u003e1992\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;45\u003c/em\u003e, 13244.\u003c/li\u003e\n \u003cli\u003eHuber K. P. Molecular spectra and molecular structure: IV. Constants of diatomic molecules. \u003cem\u003eSpringer Science \u0026amp; Business Media\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e2013\u003c/strong\u003e.\u003c/li\u003e\n \u003cli\u003eMorse M. D.; Hansen G. P.; Langridge‐Smith P. R. R.; et al. Spectroscopic studies of the jet‐cooled nickel dimer. \u003cem\u003eJ. Chem. Phys.\u003c/em\u003e \u003cstrong\u003e1984\u003c/strong\u003e, \u003cem\u003e80\u003c/em\u003e, 5400-5405.\u003c/li\u003e\n \u003cli\u003eKant A. Dissociation energies of diatomic molecules of the transition elements. I. Nickel.\u003cem\u003e\u0026nbsp;J. Chem. Phys.\u003cstrong\u003e\u0026nbsp;\u003c/strong\u003e\u003c/em\u003e\u003cstrong\u003e1964\u003c/strong\u003e, \u003cem\u003e41\u003c/em\u003e, 1872-1876.\u003c/li\u003e\n \u003cli\u003eZhang M.; Zhang J.; Gu T.; et al. First-principles investigations of chirality in trimetallic alloy clusters: AlMnAu\u003cem\u003e\u003csub\u003en\u0026nbsp;\u003c/sub\u003e\u003c/em\u003e(n=1\u0026ndash;7). \u003cem\u003eJ. Phys. Chem. A\u003cstrong\u003e\u0026nbsp;\u003c/strong\u003e\u003c/em\u003e\u003cstrong\u003e2015\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;119\u003c/em\u003e, 3458-3470.\u003c/li\u003e\n \u003cli\u003eElgavi H.; Krekeler C.; Berger R.; et al. Chirality in copper nanoalloy clusters. \u003cem\u003eJ. Phys. Chem. C\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e2011\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;116\u003c/em\u003e, 330-335.\u003c/li\u003e\n \u003cli\u003eDance I. G.; Fisher K. J. Density functional calculations of electronic structure, geometric structure and stability for molecular manganese sulfide clusters. \u003cem\u003eJ. Chem. Soc., Dalton Trans.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e1997\u003c/strong\u003e, \u003cem\u003e15\u003c/em\u003e, 2563-2576.\u003c/li\u003e\n \u003cli\u003eMing F.; Wang K.; Pan S.; et al. Assembling and disassembling Ag clusters on Si (111)-(7\u0026times;7) by vertical atomic manipulation. \u003cem\u003eACS nano\u003cstrong\u003e\u0026nbsp;\u003c/strong\u003e\u003c/em\u003e\u003cstrong\u003e2011\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;5\u003c/em\u003e, 7608-7616.\u003c/li\u003e\n \u003cli\u003eWu X.; Liang X.; Du Q.; et al. Medium-sized (n=14\u0026ndash;20) clusters: a combined study of photoelectron spectroscopy and DFT calculations. \u003cem\u003eJ. Phys-Condens Mat.\u003c/em\u003e \u003cstrong\u003e2018\u003c/strong\u003e, \u003cem\u003e30\u003c/em\u003e, 354002.\u003c/li\u003e\n \u003cli\u003eCleveland C. L.; Landman U. The energetics and structure of nickel clusters: Size dependence. \u003cem\u003eJ\u003c/em\u003e.\u003cem\u003e\u0026nbsp;Chem. Phys\u003c/em\u003e.\u003cem\u003e\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e1991\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;94\u003c/em\u003e, 7376-7396.\u003c/li\u003e\n \u003cli\u003eAlonso J. A. Electronic and atomic structure, and magnetism of transition-metal clusters. \u003cem\u003eChem. Rev.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e2000\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;100\u003c/em\u003e, 637-678.\u003c/li\u003e\n \u003cli\u003eGrigoryan V. G. Springborg M. Structural and energetic properties of nickel clusters: 2⩽N⩽\u0026nbsp;150.\u003cem\u003e\u0026nbsp;Phys. Rev. B\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e2004\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;70\u003c/em\u003e, 205415.\u003c/li\u003e\n \u003cli\u003eDeshpande M.; Kanhere D. G.; Pandey R. Structures, energetics, and magnetic properties of Ni\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eB clusters with n= 1\u0026ndash;8, 12. \u003cem\u003ePhys. Rev. A\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e2005\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;71\u003c/em\u003e, 063202.\u003c/li\u003e\n \u003cli\u003eMenon M.; Connolly J.; Lathiotakis N.; et al. Tight-binding molecular-dynamics study of transition-metal clusters. \u003cem\u003ePhys. Rev. B\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e1994\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;50\u003c/em\u003e, 8903.\u003c/li\u003e\n \u003cli\u003eJin Y.; Maroulis G.; Kuang X.; et al. Geometries, stabilities and fragmental channels of neutral and charged sulfur clusters: S\u003cem\u003e\u003csub\u003en\u003c/sub\u003e\u003c/em\u003eQ (n= 3\u0026ndash;20, Q= 0,\u0026plusmn;1). \u003cem\u003ePhys. Chem. Chem. Phys.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e2015\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;17\u003c/em\u003e, 13590-13597.\u003c/li\u003e\n \u003cli\u003eStreitwieser A. Molecular orbital theory for organic chemists. \u003cem\u003eNew York: Wiley,\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e1961\u003c/strong\u003e.\u003c/li\u003e\n \u003cli\u003eJung J.; Kim H.; Han Y. K. Can an electron-shell closing model explain the structure and stability of ligand-stabilized metal clusters?.\u003cem\u003e\u0026nbsp;J. Am. Chem. Soc.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e2011\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;133\u003c/em\u003e, 6090-6095.\u003c/li\u003e\n \u003cli\u003ePetrucci R. H.; Harwood W. S.; Herring F. G. General chemistry: principles and modern applications. \u003cem\u003ePrentice Hall,\u003c/em\u003e\u003cstrong\u003e\u0026nbsp;2002\u003c/strong\u003e.\u003c/li\u003e\n \u003cli\u003eGillespie R. J. The VSEPR model revisited. Chem. \u003cem\u003eSoc. Rev.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e1992\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;21,\u0026nbsp;\u003c/em\u003e59-69.\u003c/li\u003e\n \u003cli\u003eGillespie R. J. The valence-shell electron-pair repulsion (VSEPR) theory of directed valency.\u003cem\u003e\u0026nbsp;J. Chem. Educ.\u003c/em\u003e \u003cstrong\u003e1963\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;40,\u0026nbsp;\u003c/em\u003e295.\u003c/li\u003e\n \u003cli\u003eLeimkuhler B.; Noorizadeh E.; Theil F. A gentle stochastic thermostat for molecular dynamics. \u003cem\u003eJ. Stat. Phys.\u003c/em\u003e \u003cstrong\u003e2009\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;135\u003c/em\u003e, 261-277.\u003c/li\u003e\n \u003cli\u003eRodr\u0026iacute;guez-Carrera S.; Rodr\u0026iacute;guez-Kessler P L.; Ambriz-Vargas F. et al. First principles study for Ag-based core-shell nanoclusters with 3d-5d transition metal cores for the oxygen reduction reaction.\u003cem\u003e\u0026nbsp;Inorg. Chim. Acta\u003c/em\u003e \u003cstrong\u003e2024\u003c/strong\u003e, \u003cem\u003e572\u003c/em\u003e, 122301.\u003c/li\u003e\n \u003cli\u003eRodr\u0026iacute;guez-Kessler P. L.; Mu\u0026ntilde;oz-Castro A.; Alonso-D\u0026aacute;vila P. A. et al. Structural, electronic and catalytic properties of bimetallic Pt\u003csub\u003en\u003c/sub\u003eAg\u003csub\u003en\u003c/sub\u003e (\u003cem\u003en\u003c/em\u003e= 1\u0026ndash;7) clusters.\u0026nbsp;\u003cem\u003eJ. Alloys Compd.\u003c/em\u003e \u003cstrong\u003e2020\u003c/strong\u003e, \u003cem\u003e845\u003c/em\u003e,\u0026nbsp;155897.\u003c/li\u003e\n \u003cli\u003eRodr\u0026iacute;guez-Kessler P. L.; Rodr\u0026iacute;guez-Dom\u0026iacute;nguez A. R.; Mu\u0026ntilde;oz-Castro A. On the structure and reactivity of Pt\u003csub\u003en\u003c/sub\u003eCu\u003csub\u003en\u003c/sub\u003e(\u003cem\u003en\u003c/em\u003e= 1\u0026ndash;7) alloy clusters. \u003cem\u003ePhys. Chem. Chem. Phys.\u003c/em\u003e \u003cstrong\u003e2021\u003c/strong\u003e, \u003cem\u003e23\u003c/em\u003e(12), 7233-7239.\u003c/li\u003e\n \u003cli\u003eApsel S. E.; Emmert J. W.; Deng J. et al. Surface-enhanced magnetism in nickel clusters. \u003cem\u003ePhys. Rev. Lett.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e1996\u003c/strong\u003e,\u003cem\u003e\u0026nbsp;76\u003c/em\u003e, 1441.\u003c/li\u003e\n \u003cli\u003eSong W.; Lu W. C.; Wang C. Z.; et al. Magnetic and electronic properties of the nickel clusters Ni\u003cem\u003e\u003csub\u003en\u0026nbsp;\u003c/sub\u003e\u003c/em\u003e(n⩽30). \u003cem\u003eComput\u003c/em\u003e. \u003cem\u003eTheor\u003c/em\u003e. \u003cem\u003eChem\u003c/em\u003e. \u003cstrong\u003e2011\u003c/strong\u003e, \u003cem\u003e978\u003c/em\u003e, 41-46.\u003c/li\u003e\n \u003cli\u003eZhang M.; He L. M.; Zhao L. X.; et al. Tuning magnetic moments by 3d transition-metal-doped Au6 Clusters.\u003cem\u003e\u0026nbsp;J\u003c/em\u003e. \u003cem\u003ePhys\u003c/em\u003e. \u003cem\u003eChem\u003c/em\u003e. \u003cem\u003eC\u003c/em\u003e \u003cstrong\u003e2009\u003c/strong\u003e, \u003cem\u003e113\u003c/em\u003e, 6491-6496.\u003c/li\u003e\n \u003cli\u003eRodr\u0026iacute;guez-Kessler P. L.; Rodr\u0026iacute;guez-Carrera S.; Mu\u0026ntilde;oz-Castro A. Hydrogen evolution reaction on Ag\u003csub\u003e12\u003c/sub\u003eM icosahedrons: A DFT study. \u003cem\u003eChem. Phys. Lett.\u003c/em\u003e \u003cstrong\u003e2024\u003c/strong\u003e, \u003cem\u003e856\u003c/em\u003e,\u0026nbsp;141588.\u003c/li\u003e\n \u003cli\u003eOlalde-L\u0026oacute;pez D.; Rodr\u0026iacute;guez-Kessler P. L.; Rodr\u0026iacute;guez-Carrera S.; et al. Hydrogen storage properties for bimetallic doped boron clusters M\u003csub\u003e2\u003c/sub\u003eB\u003csub\u003e7\u003c/sub\u003e (M= Fe, Co, Ni). \u003cem\u003eInt. J. Hydrogen Energy\u003c/em\u003e\u003cem\u003e.\u003c/em\u003e 2024.\u003c/li\u003e\n \u003cli\u003eJana G.; Pan S.; Rodr\u0026iacute;guez-Kessler P. L.; et al. Adsorption of molecular hydrogen on lithium\u0026ndash;phosphorus double-helices. \u003cem\u003eJ. Phys. Chem. C\u003c/em\u003e \u003cstrong\u003e2018\u003c/strong\u003e, \u003cem\u003e122\u003c/em\u003e(49),\u0026nbsp;27941-27946.\u003c/li\u003e\n \u003cli\u003eRodr\u0026iacute;guez-Kessler P. L.; Murillo F.; Rodr\u0026iacute;guez-Dom\u0026iacute;nguez A. R.; et al. Structure of V-doped Pdn (n= 2\u0026ndash;12) clusters and their ability for H\u003csub\u003e2\u003c/sub\u003e dissociation. \u003cem\u003eInt. J. Hydrogen Energy\u003c/em\u003e\u003cem\u003e.\u003c/em\u003e \u003cstrong\u003e2018\u003c/strong\u003e, \u003cem\u003e43\u003c/em\u003e(45), 20636-20644.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Cluster structure, hydrogen storage, DFT investigation","lastPublishedDoi":"10.21203/rs.3.rs-5906730/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-5906730/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eComputational cluster sciences are rooted in geometrical optimization successes of small groups of atoms or molecules. Following size growth and element increases, optimizations are arduous and hardly generalized in certain patterns, despite advances in calculation algorithm and computing powers. Herein, a disassembly-assembly strategy is introduced to reach stable structures of binary Ni\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eS\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e (\u003cem\u003en\u003c/em\u003e = 2-10) clusters. The lowest-energy Ni\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003eS\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e isomers can be viewed as nestifications of low-lying Ni\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e and S\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e components. Identical spatial orientations in and out of the Ni-S binary systems are kept for the elemental S\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e and Ni\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e clusters. At the same corresponding number \u003cem\u003en\u003c/em\u003e, the metal and chalcogenide clusters coincidentally possess the same point group and such a trend is kept up to the NiS crystalline phase. As a result, the growth pattern of the compound clusters is simply viewed as an assembly process of their component clusters. Furthermore, molecular orbital analysis indicates that the nested structures are endowed with a large amount of dangling bonds to adsorb hydrogen atoms for hydrogen storage. Such an assembly-disassembly route in structural optimizations is hoped to serve as a route map in future functional cluster predictions.\u003c/p\u003e","manuscriptTitle":"Structural disassembly, dangling bond and hydrogen storage of NinSn (n=2-10) clusters","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-04-29 10:38:12","doi":"10.21203/rs.3.rs-5906730/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"8f1ee9e4-f31c-4aa6-836e-a254547066ec","owner":[],"postedDate":"April 29th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2025-11-06T12:38:13+00:00","versionOfRecord":[],"versionCreatedAt":"2025-04-29 10:38:12","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-5906730","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-5906730","identity":"rs-5906730","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}