Zinc is a group-IIB element that is frequently used as a galvanizing material and in storage media as an anode . However, its ability to lose electrons quickly to oxygen makes it unsuitable as a coating material. Zinc exhibits a s2 closed-shell structure, and its dimer forms through van der Waal (vdW) forces . As the cluster size increases, the properties of the clusters change significantly, and the effect of vdW forces decreases. Bulk zinc has metallic characteristics because of the overlap of the s and p orbitals. In the past, Zn clusters have been analyzed both experimentally and theoretically, where the studies were mainly conducted to determine the stability and electronic properties of the zinc cluster ground state . The majority of the research work on Zn clusters is focused on the vdW transition for the Zn clusters. For example, Wang et al. investigated it by using the PW91 functional, which suggested that the transition starts from n = 8. Iokibe et al. obtained a similar result using density functional theory (DFT) calculations at different levels of theory to study the transition states (vdW to semiconductor-like states) in Zn clusters. In addition, the approaches used to study the electronic properties, such as ionization potentials (IPs) of zinc, are based on the ∆-SCF methods; for some clusters, such as Zn2, the results significantly underestimate the experimentally measured IPs. State-of-the-art approaches, such as GW approximation, have been proven to provide accurate IPs and electron affinity (EA) values for various clusters .

Determining the ground states of clusters is essential; several metastable isomers are present in an experimental study, which can introduce difficulties in determining the ground state structure. Several algorithms have been used to describe the ground state and low-lying structures of the clusters. Among these approaches, particle swarm optimization (PSO), combined with density functional approximations, was used to determine the ground state structure. Thus, one can efficiently locate the global minimum in the potential energy surface. Based on the PSO algorithm, Wang et al. developed a code called CALYPSO (“Crystal Structure Analysis by Particle Swarm Optimization”). It has been used previously by many researchers in discovering new materials . In addition to the ground state properties, electronic properties such as ionization energies (IEs) and HOMO–LUMO gaps are also important, as they determine the physical properties of the clusters. Previous attempts to study the electronic properties of Zn clusters based on ∆-SCF methods tended to underestimate the ionization energies of the clusters as the size grew. State-of-the-art techniques, such as the GW method, can effectively describe the electronic properties of many clusters with higher accuracy. In addition to G0W0, we have also applied the correction. The unavailability of experimental data is a key issue because of which the determination of ground state and electronic properties remains unexplored. To our knowledge, no G0W0 studies on neutral zinc clusters have been reported in the literature. Our G0W0 calculations will provide a benchmark to help accelerate the research on clusters and creating materials with high stability that can be used for advanced energy storage applications .

In this work, we have employed the generalized gradient approximation (GGA) to optimize the results and to obtain the ground state structures and the isomers of neutral Zn clusters. Furthermore, we have performed G0W0 calculations using the FHI-aims all-electron code to study electronic properties such as IPs, electron affinities (EAs), and HOMO–LUMO gaps of Zn clusters. In addition, the study of low-lying isomers is also carried out to compare the metastable structures with the ground state, which ensures that the obtained lowest-energy structure is the actual ground state.

All geometric optimization calculations were carried out with the PBE exchange–correlation functional of the GGA. The structure prediction in our work was carried out by the CALYPSO code with ABACUS software for structure optimization . The non-relativistic ONCV-type pseudopotential (SG15 V1.0) was used. The obtained structures have been carefully analyzed with the VESTA software, and low-energy isomers were refined from more than 600 structures (ca. 22 generations in CALYPSO).

The geometric optimization of all clusters for a size range of n = 2–15 was performed in two steps: (i) structure search and initial geometric optimization, within which the distinct structures were separated using the GGA (PBE) functional, and (ii) high-precision optimization by ABACUS using the “accurate” setting. PBE instead of the hybrid functional was used because of its low cost; also, it brings a non-empirical functional that can easily and reliably predict new systems and properties. The next step was to perform the G0W0 calculations using the FHI-aims all-electron package , which was further used to assess the electronic properties of the Zn clusters. We used the pre-relaxed structures to obtain the energy gaps and IPs of the Zn clusters. In FHI-aims, G0W0 calculations employing the NAO basis sets were performed with the PBE functional to relax the structure with “tier 4” and “tight” settings. The results obtained from G0W0 calculations predict better IPs and energy gaps of the molecules and clusters. The results are compared with the previously available experimental and theoretical data to validate our work.

In this section, we will present the results of the structural relaxation, stability, and electronic properties of neutral Zn clusters. The predictions of the various geometrical structures are presented, and binding energies are discussed. In addition, we have also explored the ionization potentials, electron affinities, and energy gaps for the series of Zn clusters.

Figure 1: Geometrical structures of ground state Zn clusters.

The Zn clusters show planar geometrical structures for n = 3–4. The geometrical structure of Zn8, a magic cluster obtained in this work, is similar to that of Chaves and co-workers . The octamer of zinc is particularly interesting, as the transition from vdW forces to metallic bonds occurs at n = 8. However, the geometrical structure of the ground state is still controversial. Recently, Chaves et al. studied various transition element clusters and found a new ground state of the zinc cluster with n = 8. The zinc cluster with nine atoms was also obtained by Iokibe et al. , who predicted the Zn10 structure obtained in our work. They used the PW91 functional to calculate the binding energies of medium-sized Zn clusters. The cluster geometries from n = 11–15 were also predicted by Li et al. using the B88 functional. The structure of Zn14 was also obtained as the second lowest in our study and by Wang et al. . This structure was reported as the lowest-energy structure.

In order to study the stability of the generated structures, it is essential to determine the binding energy per atom of each cluster isomer, which can be defined as:

where Etot is the total energy of the cluster after relaxation, n is the size of the cluster, and Eatom is the energy of a free atom. Here, we have also employed spin-polarized calculations to obtain the binding energies of Zn clusters. For metallic systems, spin effects significantly influence the total energies, and neglecting these effects can result in overestimating binding energies. The dimer binding energy obtained in our work is 0.022 eV, close to the experimental value of 0.03 eV . We have also calculated the binding energy of the zinc dimer by using the FHI-aims code and PBE relaxed calculations. The output geometries from CALYPSO and ABACUS relaxed calculations are used as input, and the PBE relaxed calculation is performed in the FHI-aims code. The obtained binding energy from the FHI-aims code for the dimer is 0.034 eV, the same as the experimental binding energy. It should be noted that the lower binding energy of zinc dimers is mainly due to the weak vdW bonding effects .

Figure 2: Binding energy per atom (eV) of zinc cluster ground states for the size range n = 2–15 compared with experimental and theoretical works .

As zinc has two valence electrons, these peaks correspond to the formation of magic number clusters. Eight and 20 valence electrons make a cluster more stable, which is seen as a peak in the binding energy curve of Zn clusters. For cluster sizes of n = 11–15, Eb increases smoothly. Our predicted binding energies are similar to those from an earlier study by Iokibe et al. and Wang et al. , who determined the stability and structural properties of neutral Zn clusters by employing the PW91 functional. Analyzing the performance of different functionals for Zn clusters, it can be concluded that the LDA overestimates the binding energy significantly. Our PBE results are in good agreement with the PW91 results reported by Iokibe et al. and Wang and co-workers .

Figure 3: Ionization potentials of Zn clusters for the size range n = 2–15. Comparison of our work G0W0@PBE with PW91 , ∆SCF , and LDA .

Table 1: Electron affinity, Egap, and hardness for neutral clusters of zinc.

aRef. PW91 ; bRef. LDA .

Figure 4: Comparison of Egap of neutral Zn clusters (n = 2–15) from G0W0 and calculations.

The ground state and electronic properties of zinc clusters for sizes n = 2–15 are studied in this work. Structures of the Znn clusters have been generated using the CALYPSO code (interfaced with ABACUS) using the PBE functional. The binding energies show the two-knee behavior also seen in metal clusters of various species. Electronic properties, such as IP, EA, and the relative hardness of the clusters, have been obtained using the G0W0@PBE in FHI-aims. The IP benchmark shows the size evolution behavior of the Zn clusters. The benchmark results are in good agreement with previously reported data. Moreover, the corrections showed substantially improved ionization potential values compared to the G0W0 approach.