Three-Dimensional Assignment of the Structures of Atomic Clusters: an Example of Au8M (M=Si, Ge, Sn) Anion Clusters

Identification of different isomer structures of atomic and molecular clusters has long been a challenging task in the field of cluster science. Here we present a three-dimensional (3D) assignment method, combining the energy (1D) and simulated (2D) spectra to assure the assignment of the global minimum structure. This method is more accurate and convenient than traditional methods, which only consider the total energy and first vertical detachment energies (VDEs) of anion clusters. There are two prerequisites when the 3D assignment method is ultilized. First, a reliable global minimum search algorithm is necessary to explore enough valleys on the potential energy surface. Second, trustworthy simulated spectra are necessary, that is to say, spectra that are in quantitative agreement. In this paper, we demonstrate the validity of the 3D assignment method using Au8M− (M = Si, Ge, Sn) systems. Results from this study indicate that the global minimum structures of Au8Ge− and Au8Sn− clusters are different from those described in previous studies.

Nanoclusters, including small groups of atoms or molecules, possess an intermediate size range between single atoms and condensed matter. Owing to the unique properties of the clusters in designing new types of nanofunctional materials, many efforts have been devoted to demonstrating that the properties of the clusters depend on their structure and composition 1,2 . Once low-lying structures are found, their properties can be calculated and compared with measured values to make precise structural assignments. Therefore, it is very important to determine the exact structure and composition of these nanoclusters 3,4 .
In the past two decades, various techniques have been used to investigate the structures of nanoclusters, such as photoelectron spectroscopy [5][6][7][8][9][10] , ion mobility 11,12 , infrared multiphoton dissociation spectroscopy 13,14 , electron diffraction 15 , X-ray diffraction 16 , coulomb explosion [17][18][19] , and trapped ion electron diffraction 20 . All of these techniques are quite powerful for obtaining structural information when combined with optimization algorithms and density functional theory (DFT) calculations. So far, the different types of algorithms have been developed to search the low-energy structure on the complicated potential energy surface (PES), such as genetic algorithms (GA) 21 , simulated annealing (SA) 22 , and basin hopping (BH) [23][24][25][26] . These experimental techniques and optimization algorithms are very useful for finding the low-energy structure on PES. However, the assignment of cluster structures remain to be difficult, specifically identifying the structural and energetic close isomers 27 . Therefore, determining the different isomers and confirming which structure is the global minimal structure in low-lying structures is still a fundamental problem in atomic clusters.
The conventional method used to distinguish the global minimum from other isomers is to calculate their relative energies and compare the calculated vertical detachment energies (VDEs) with the experimental value 28 . This methodology is not very effective at recognizing the lowest energy structure and the energetically close isomers 27 because the relative energy calculations depend on the method and basis set. The different method and basis set can lead to a change of energy order. On the other hand, normally the vertical detachment energy differences of different isomers, especially for the low energy isomers, are very small. Therefore, it is difficult to distinguish the lowest energy structure from different isomers only by the relative energy and VDE values. To more reliably identify the global minimum from all of the isomers, we present a three-dimensional (3D), including energy (1D) and density of state (DOS) (2D), assignment method for effectively analyzing different low-energy structures of anion clusters. The simulated DOS spectra are based on a generalized Koopmann's theorem 29 . It can effectively simulate the photoelectron spectra obtained by experiment to make the structure assignment. We used Au 8 M − (M = Si, Ge, Sn) as an example system to illustrate the validity of our method. We then compared the DOS spectra for the isomers of Au 8 M − (M = Si, Ge, Sn) systems with previous studies, thereby providing considerable credence for the identified isomers of these clusters.

Results
To determine the global minima of the Au 8 M − (M = Si, Ge, Sn) systems, we searched more than 200 isomeric forms using the BH method 23 Table 1 Figure S1 in the Supporting Information. The experimental spectra of the Au 8 M − (M = Si, Ge, Sn) systems are showed in Fig. 2a~c. Table 1 shows that the results calculated from different theoretical levels lead to different energy values. Therefore, it is difficult for us to distinguish which isomer is the global minimum by their total energy and first VDEs. To accurately obtain the lowest energy structure, we compared their total energy and DOS spectra with the experimental results. The simulated spectrum of isomer 1 for the Au 8 Si − cluster agrees well with the experimental data ( Fig. 2a) obtained by Wang, et al. 31 and should be the lowest-energy structure by our calculation. Due to the spin-orbit effects included for the Au atom, our   simulated spectrum is better than previous studies (Fig. 2d) 31 , which had not considered the spin-orbit effects for the Au atom. The experimental spectra (Fig. 2b,c) of Au 8 Ge − and Au 8 Sn − clusters are very similar to the literature 31 , which suggests that their primary structures should be similar with each other. Based on the comparison of the total energy and DOS spectra of isomers 1 and 2 for the Au 8 Sn − cluster, we confirmed that the primary structure of Au 8 Sn − should be isomer 2. The simulated spectrum of isomer 2 agrees well with the experimental spectra (Fig. 2c) and suggests that isomer 2 is more likely to exist under real conditions. Comparing experimental and theoretical spectra for the Au 8 Ge − cluster 31 , the simulated spectrum of isomer 4 (Figure. S1i in the Supporting Information) together with isomer 3 (Figure. S1h in the Supporting Information) has good agreement with the experimental spectrum (Fig. 2b). However, isomer 4 has a relatively higher energy than the other structures at several different theoretical levels (Table 1). Due to the similar experimental spectra of the Au 8 Ge − and Au 8 Sn − clusters, we believe that isomer 4 should be a primary structure of the Au 8 Ge − cluster. To further verify those structures analyzed by our method, the distributions of conformational populations were calculated at the MP2/ Def2-TZVPPD level of theory (using Gaussian09 software package, revision D.02, Gaussian, Inc.), and the results are summarized in Table 2. The conformational populations depending on temperature can be found in Figure S2. The results show those primary structures of the Au 8 M − (M = Si, Ge, Sn) system obtained by our method hold a high percentage in the range of 1 to 500 K. For the structure analysis, the contrast of multi-dimensional characters, including energy, PES, and infrared spectra, can more accurately distinguish the different isomer configurations and reduce the uncertainty of the structure assignment. Through structure searching, the global minimum structure of M N+1 usually can be found from one of the low-lying isomers of M N using our previous calculations [32][33][34] . Therefore, we have speculated that some low-lying isomers of the Au 8 Ge − and Au 8 Sn − systems may coexist under certain experimental conditions. Two possible evolution routes are presented in Fig. 3 for the Au 8 Ge − and Au 8 Sn − systems. The structure evolution routes provide us with another method to analyze the global minimum structure.

Discussion
In this study, we have presented a three-dimensional structural assignment method based on energies and DOS spectra to distinguish different isomers. The method first involves exploring the PES using an optimization algorithm and obtaining sufficient isomers. Secondly, the DOS spectra of the low energy structures were simulated based on a generalized Koopmann's theorem. The three-dimensional characteristics of cluster structure can be seen as the fingerprint of different isomers. This method will be more effective to determine which structures exist under real conditions. Applying the method to the Au 8 M − (M = Si, Ge, Sn) systems, we found that the primary structures of Au 8 Ge − and Au 8 Sn − cluster are different from previous studies. The three-dimensional structure assignment method is simple and effective for different types of clusters to distinguish their structures.

Methods
The basin hopping (BH) algorithm combined with density functional theory has been used to search the potential energy surface (PES). Generalized gradient approximation in the Perdue-Burke-Ernzerhof isomers were re-optimized using the functional PBEPBE and a scalar relativistic effective core potential Stuttgart/Dresden (SDD) basis set for all of elements in the Gaussian 09 software package (revision D.02, Gaussian, Inc.). The DOS spectra for all candidate isomers were calculated using PBE0 functional and CRENBL basis set for Si, Ge, and Sn, CRENBL basis set for Au with spin-orbit effects included in the NWChem software package 30 . The DOS spectra were calculated using the following steps: (i) calculate the first vertical detachment energies (VDEs) of anion clusters, which was defined as the energy difference between the optimized anion isomer and the neutral at the corresponding anion geometry, (ii) add the binding energies of deeper orbitals of the anion cluster to the first VDEs to approximate the higher binding energy detachment features, and (iii) fit each peak with a 35-meV-wide Gaussian curve. Each peak can be seen as a molecular orbital. Therefore, we can obtain the approximate electronic structures of the anion cluster using this method. Because each anion cluster has a unique electronic structure, we can easily distinguish the difference of each isomer by their DOS spectra. This method is very effective for anion clusters, and we used this method to study the pure Au or doped-Au anion clusters in previous studies 33,34,[36][37][38][39][40] .     Table 1.