Abstract
The structural and electronic properties for the global minimum structures of mediumsized neutral, anionic and cationic Si_{n}^{μ} (n = 20–30, μ = 0, −1 and +1) clusters have been studied using an unbiased CALYPSO structure searching method in conjunction with firstprinciples calculations. A large number of lowlying isomers are optimized at the B3PW91/6311 + G* level of theory. Harmonic vibrational analysis has been performed to assure that the optimized geometries are stable. The growth behaviors clearly indicate that a structural transition from the prolate to sphericallike geometries occurs at n = 26 for neutral silicon clusters, n = 27 for anions and n = 25 for cations. These results are in good agreement with the available experimental and theoretical predicted findings. In addition, no significant structural differences are observed between the neutral and cation charged silicon clusters with n = 20–24, both of them favor prolate structures. The HOMOLUMO gaps and vertical ionization potential patterns indicate that Si_{22} is the most chemical stable cluster and its dynamical stability is deeply discussed by the vibrational spectra calculations.
Introduction
The experimental and theoretical studies of the atomic and molecular clusters are interesting topics since they constitute intermediate phases between individual atoms and bulk solids, which can be used to understand how the fundamental properties of materials evolve from isolated atoms or small molecules to a bulk phase^{1,2,3,4,5,6,7,8}. The study of small clusters can help us to design better nanosystems with specific physical and chemical properties. Silicon is the most widely used material in the microelectronic industry. If current miniaturization trends continue, minimum device features will soon approach the size of atomic clusters. In this size regime, the structures and properties of materials often differ dramatically from those of the bulk. The study of the structures and properties of silicon clusters has been an extremely active area of current research. During the past two decades, a large number of experimental and theoretical studies have been carried out in this direction^{4,6,7,8,9,10}. Much attention has been focused on understanding the structural and growth behavior of small or mediumsized silicon clusters^{4,10,11,12,13,14}.
Several highresolution photoelectron, Raman and infrared spectra experiments have been carried out to understand the atomic structure of small silicon clusters and showed that both Si_{6} and Si_{10} have exceptional stability^{11}. Ion mobility measurements have revealed much of what is known about the growth behaviors of mediumsized silicon clusters^{14}. Jarrold et al. have determined that anionic silicon clusters are prolate shape for n < 27 and become more sphericallike geometry for larger clusters^{11}. However, the transition from prolate to more sphericallike geometries for cationic silicon clusters was observed in between 24 < n < 30^{11,12,13,14}. Up to now, most of spherical and compact clusters have been considered as theoretical models attempting to support this measurement. A lot of works have been carried out with results not always in agreement between authors^{14,15,16,17,18,19,20}. Despite the enormous progress that has been made, the true lowestenergy structures for the silicon clusters in the size range of 20 ≤ n ≤ 30 are still debatable. The main reasons may be as follows: (i) The procedure used in the case of small clusters is not practical for larger clusters. (ii) The predicted global minima are subtle sensitivity for the selected density functional theory, or the molecularorbital level in the ab initio calculations. Moreover, the determination of the true global minimum structure is also a challenging problem, because of the much increased complexity of the potential surface as well as the exponential increase of the lowestenergy structure with the number of atoms in the cluster^{21}.
In order to systematically study the structural evolution and electronic properties of silicon clusters, we here present extensive structure searches to explore the global minimum geometric structures of mediumsized neutral and charged silicon clusters in the size range of 20 ≤ n ≤ 30, by combining our developed CALYPSO method with the density functional theory. Our first goal of this work is to gain a fundamental understanding of the ground state geometric structures in mediumsized silicon clusters. The second one is to reexamine a number of neutral and charged lowenergy isomers of Si_{20}Si_{30} that have been reported previously by experiments or density functional calculations. Thirdly, we are also motivated to explore the physical mechanism of the growth behaviors of mediumsized silicon clusters and provide relevant information for further theoretical and experimental studies. The paper is organized as follows: the computational details are described in Section 2, results are presented and discussed in Section 3 and our final conclusions are given in Section 4.
Computational Methods
Our cluster structure prediction is based on the CALYPSO method^{22,23,24}. A global version of particle swarm optimization (PSO) algorithm^{25} is implemented to utilize a fine exploration of potential energy surface for a given nonperiodic system. The bond characterization matrix (BCM) technique is employed to eliminate similar structures and define the desirable local search spaces. This structure prediction method has been benchmarked on LJ clusters with sizes up to 150 atoms. High search efficiency is achieved, demonstrating the reliability of the current method. The significant feature of this method is the capability of predicting the stable structure with only the knowledge of the chemical composition. It has been successful in correctly predicting structures for various systems^{24,25,26,27}. The evolutionary variable structure predictions of the neutral and charged Si_{20–30} clusters are performed. To seek lowlying structures, the computational process can be divided into two steps. Firstly, an unbiased global search is performed, using the CALYPSO method combined with density functional theory geometric optimization. Each generation contain 30 structures, 60% of which are generated by PSO. The others are new and will be generated randomly. We follow 50 generations to achieve the converged structure. Next, among the 1000–1500 isomers for the neutral and charged Si_{n} clusters, the top fifteen lowlying isomers are collected as candidates for the lowestenergy structure. Those isomers with energy difference from the lowestlying isomer less than 3 eV are further optimized at B3PW91/6311 + G* level of functional/basis set. All the quantum chemical calculations are performed using the Gaussian 09 program package^{28}. The convergence thresholds of the maximum force, rootmeansquare (RMS) force, maximum displacement of atoms and RMS displacement are set to 0.00045, 0.0003, 0.0018 and 0.0012 a.u., respectively. The effect of the spin multiplicity is also taken into account in the geometric optimization procedure. Meanwhile, the vibrational frequency calculations are performed at the same level of theory to make sure that the structures correspond to real local minima without imaginary frequency.
To verify the reliability of our calculations, we have calculated the neutral and charged silicon dimers (Si_{2}, Si_{2}^{–} and Si_{2}^{+}) through many different functionals (HF^{29}, MP2^{30}, B3LYP^{31,32}, PW91^{31,33}, PBE^{34}, B3PW91^{31,33,35}, B3P86^{32} as well as CCSD(T)^{36}) with 6–311 + G* basis sets. The calculated results are summarized in Table 1. From Table 1, it is found that the results of bond length (r) and vibrational frequency (ω_{e}) of the threetype silicon dimers based on both B3PW91 and CCSD(T) methods are in good agreement with the experimental values^{37,38,39,40}. While the calculated dissociation energy (D_{e}) of neutral Si_{2}, adiabatic electronic affinity energy (AEA) of anionic Si_{2}^{–} and adiabatic ionization potential (AIP) of cation Si_{2}^{+} at B3PW91 level of theory are closer to the experimental values^{37,40}, with deviation less than 2%, 5% and 3%, respectively. So, the B3PW91/6311 + G* has been selected as the reasonable method for silicon clusters.
Results and Discussion
The structures found by CALYPSO searches in the range from 20 to 30 can be categorized into two kinds: prolate and sphericallike structures. All earlier known structures, experimentally and theoretically, were successfully reproduced and optimized in our current structure searches. Here, we only selected the lowest energy structures and the second lowlying isomers for neutral, anionic and cationic species and displayed them in Figs 1, 2 and 3, respectively. The other lowlying isomers of the threetype silicon clusters together with their relative energies are presented in Figures S1, S2 and S3 (see Electronic Supplementary Information). To further confirm the reliability of the present computational method, the vertical detachment energies (VDEs), adiabatic detachment energies (ADEs) and vertical ionization energies (VIPs) for large anionic and neutral silicon clusters are also calculated. The theoretical results as well as the experimental data are listed in Table 2^{41}. The agreement between the experimental data and the calculated results is also excellent, which shows the accuracy of the present theoretical calculations.
Geometric structures
In order to gain information on the growth of silicon clusters, many attempts have also been made to study the geometries of lowlying mediumsized neutral and charged silicon clusters. These include injectedion drifttube techniques, photoelectron spectroscopy measurements and ab initio calculations^{11,18,42,43,44,45}. For neutral silicon clusters, our results indicate that the prolate structures are more stable than sphericallike structures for Si_{n} (20 ≤ n ≤ 25) clusters, then a structural transition to more sphericallike structure occurs at n = 26. These prolate structures can be described as stacks of stable subunits. Take the Si_{20} and Si_{21} clusters for example, their structures consist of Si_{6} unit joined by other atoms. Our result on structural transition is in agreement with the cationic mobility experiment^{46}, which has shown that a structural transition from prolate to more sphericallike structures may occur in between 24 < n < 34. In addition, the previous theoretical studies on the mediumsized silicon clusters^{6,47,48} also indicated that the prolate structures are more favorable for Si_{n} (n = 20–26) clusters. In other words, the sphericallike isomers are expected to become more competitive energetically than the prolate isomers for larger Si_{n} (n ≥ 27) clusters.
Although considerable studies have been carried out for the neutral silicon clusters, only a few studies are available for charged clusters^{11,41,48}. For anionic silicon clusters, we have examined a number of lowenergy isomers which are obtained by our structural searches. Interestingly, a clear qualitative change in the geometry of these isomers is found except the silicon clusters with n ≥ 27. Further geometrical optimization for the final structures confirmed that the prolate Si_{n}^{−} isomers becomes slightly more stable than sphericallike isomers and a structural transition from prolate to more sphericallike geometries occurs at n = 27. This observation is in complete agreement with the photoelectron spectroscopy experiments and firstprinciples densityfunctional studies by Bai et al.^{40}. In order to gain insight into the electronic properties of the mediumsized charged silicon clusters, the vertical detachment energies (VDEs) and adiabatic detachment energies (ADEs) of Si_{n}^{−} (n = 20–30) are calculated. The theoretical results are listed in Table 2 together with available experimental values^{41}. It can be seen from Table 2 that the calculated VDE values of Si_{n}^{−} (n = 20–30) clusters are in good agreement with experimental values, with discrepancy in the range of 0.3% to 2.7%. In addition, we also simulated the photoelectron spectra of Si_{n}^{−} (n = 20–30) clusters and compared with the experiments^{49}. The simulated results together with experimental photoelectron spectra are shown in Figure S4 of supplementary information. It can be seen from Figure S4, the positions and the general shape of the peaks overall agree well with experimental results. These results further give us confidence in the obtained groundstate structures for these anionic clusters. However, there is no any available experimental data to compare with our obtained ADE results for Si_{n}^{−} (n = 20–30) clusters. We hope that our results for Si_{n}^{−} (n = 20–30) clusters would provide more information for further investigation in the future.
Previous mobility measurement^{45} has been carried out for Si_{n}^{+} (n = 20–27), which can provide information on the general shape and initial geometry of clusters. This measurement result shows that the cationic silicon clusters become sphericallike structures occurring in between 24 < n < 34. Based on the unbiased global search, the prolate structures (as shown in Fig. 3) are tested to be the ground state structures for Si_{n}^{+} (n = 20–24) clusters. This result mirror well the shape transition observed in mobility measurement. In addition, the theoretical study^{48} on Si_{n}^{+} (n = 20–27) clusters also reveals that compact Si_{n}^{+} structures lie above the prolate for n ≤ 23, closely compete with them for n = 24 and 25 and overtake them for n ≥ 26 in energy. It worth mentioning that no significant difference is observed between the neutral and cation charged silicon clusters with n = 20–24 (see Figs 1 and 3). For Si_{n}^{+} (n = 25–30) clusters, the sphericallike structures are more stable than prolate structures. These sphericallike geometries have neither the diamondlike packing of bulk silicon, nor the stuffed fullerene structure with an outer shell of pentagons and hexagons. For example, we find that the compact structure of Si_{22}^{+} includes the tricapped trigonal prism (TTP) Si_{9} units which are believed to appear in the prolate structures.
Considering the structural transition point from prolate to more sphericallike geometries may relate to the chosen functional, here we reoptimized the most stable prolate structures and the next lowlying isomer with sphericallike structures for three type (neutral, anion and cation) silicon clusters at the PBE level of theory. The calculated relative energies between the most stable prolate structure and the lowlying sphericallike structure at PBE level are given in parenthesis in Figs 1, 2 and 3. It can be clearly seen that although the prolate and sphericallike structures of Si_{23,25} are almost degenerated in energy at PBE level of theory, the lowest energy structures remain unchanged for neutral, anionic and cationic silicon clusters. This suggests that the ground state structures of neutral and charged silicon clusters are independence with the used functional. Interestingly, we have also found that all the lowestenergy structures favor the low spin state.
Relative stability
It is wellknown that the binding energy (E_{b}) of a given cluster is a measure of its thermodynamic stability. It is defined as the difference between the energy sum of all the free atoms constituting the cluster and the total energy of the cluster. The binding energies per atom for mediumsized neutral, anionic and cationic silicon clusters (n = 20–30) are summarized in Table S1 of the supplementary material. Meanwhile, the binding energies as a function of cluster size n are plotted in Fig. 4(a–c), respectively. As is shown in Fig. 4, all the E_{b} values are not obviously lower than that of the silicon crystal (4.75 eV)^{50}. In addition, the binding energies do not show a dependent behavior on the cluster size, which is in agreement with the experimental reports^{16,17}. This shows that the structures of silicon clusters (n = 20–30) have different growth pattern. It can be seen from Fig. 4(a) that the binding energies of prolate structures are larger than those of sphericallike types for Si_{n} (n = 20–25), indicating that the prolate structure become more competitive energetically than the nearspherical isomers. The prolate structure of Si_{25} cluster is almost as stable as the compact structure. Furthermore, it is found that the binding energies per atom for studied silicon clusters irrespective of prolate and sphericallike structures change in a narrow region of 3.48–3.58 eV, which is also confirmed by the experiment^{17}. The maximum value of 3.58 eV is found at Si_{24} with prolate structure, as well as at Si_{26} and Si_{29} clusters with sphericallike structures, which exhibits that these clusters are the most stable cluster in present study. Moreover, it is found that our calculated E_{b} (3.57 eV) of Si_{30} agree well with the previous value (3.796 eV)^{19}. The E_{b} of anionic clusters as a function of cluster size n is displayed in Fig. 5(b). From Table S1, it is found that the sphericallike and prolate Si_{20}^{−}, Si_{24}^{−} and Si_{26}^{−} clusters are degenerate in binding energies, which can also be clearly seen from their relative energy difference as shown in Fig. 4(b). For the other clusters, the E_{b} of prolate structures are higher than those of the sphericallike structures. That is to say, the prolate structures are more energetically favorable than compact sphericallike structures. The Si_{29}^{−} is the most stable structure among the obtained anionic clusters due to its largest binding energy. As for the cationic species (see Fig. 4(c)), we can clearly see that the prolate structures are more stable than the compact sphericallike structures for Si_{n}^{+} (n = 20–24) clusters. This is in agreement with the result of their relative energy order. The binding energies for the prolate and compact structures increase slightly when n is smaller than 24. Furthermore, in the sphericallike structures, a sharply increasing is found at n = 29 and up to a maximum of 3.84 eV at n = 30. Namely, the Si_{30}^{+} cluster is the most stable cluster within the cation charged clusters in the range of cluster size n = 20–30. The small cluster Si_{20}^{+} is less stable compared with the other clusters for both prolate and sphericallike structures. From the above discussions, it is clear that the transition point from prolate structures to compact sphericallike structures occurs at n = 26 for neutral silicon clusters, at n = 27 for anions and n = 25 for cations. Therefore, the accepting or loss of an extra electron strongly affects the structures of silicon clusters.
The HOMOLUMO energy gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) is a useful quantity for examining the kinetic stability. A large energy gap corresponds to a high energy required for electron excitation. The sizedependent energy gaps of the most stable Si_{n}^{μ} (n = 20–30, μ = 0, −1 and +1) clusters are summarized in Table S1 and plotted in Fig. 5. By the comparison between the energy gaps of neutral, anionic and cationic silicon clusters, we can note that the curves of energy gaps for both anionic and cationic clusters have approximate tendency when n ≥ 25. A distinct maximum occurs at charged Si_{26}^{+/−} clusters among the cluster size n = 25–30, indicating that Si_{26}^{+/−} is relatively more chemical stable in the electronic structure compared other clusters. However, within the whole studied anionic clusters, Si_{21}ˉ is the most chemical stable cluster. For neutral Si_{n} (n = 20–30) clusters, the HOMOLUMO energy gap results reveal that the Si_{22} is the the highest kinetic stability cluster with the largest HOMOLUMO gap of 2.85 eV.
In order to further check the dynamical stability of Si_{22}, Si_{21}^{−} and Si_{26}^{+} clusters, we have calculated their vibrational spectra. The infrared and Raman spectra of Si_{22} are shown in Fig. 6 and the spectra of Si_{21}^{−} and Si_{26}^{+} clusters are shown in Figure S5 of the supporting information. We have also shown the direction of motion of the ions for the frequency with the highest Raman activity or infrared intensity. In the following, we will take the infrared and Raman spectra of Si_{22} as an example to describe the dynamical stability of the neutral silicon clusters. From the insets in Fig. 6, it can be seen that Si atom localized at the outside mainly contributes to the highest peaks of infrared spectrum and Raman activity. These spectra can provide a spectroscopic fingerprint to assist experimentalists to distinguish different species and different isomers. The infrared spectra and Raman activity of Si_{22} have several peaks due to its low C_{s}symmetry. The highest intensity peak of the infrared spectra is 95.02 km/mol and the highest Raman activity is 9.14 Å^{4}/amu. They are located at 378.00 and 314.00 cm^{−1}, respectively. The complex nature of the farinfrared region when combined with the highest intensity peak of the infrared spectra can be used as a fingerprint for identifying different Si_{22} isomer. Raman activity mainly corresponds to the breathing modes and in these modes all the ions in clusters having high symmetry move together. The highfrequency peak for Raman spectra reflects the strong bonding of cluster Si_{22}.
The vertical ionization potential (VIP) is also an important parameter to assess the chemical stability of clusters. The VIP is defined as the energy differences between the total energy of neutral and cationic clusters with the same structure of the lowestenergy neutral state. Large VIPs indicate high chemical inertness. We have calculated the VIPs of the lowestenergy structures for the neutral silicon clusters. In addition, we have also calculated their adiabatic ionization potential (AIP), which is given by the formula AIP = E_{optimized cation} − E_{optimized neutral}. The calculated results are listed in Table 2. As shown in Table 2, our theoretical AIPs results of Si_{n} (n = 20–30) clusters are in good agreement with the experimental data^{18}. Si_{22} has the largest vertical ionization potential (7.28 eV), corresponding to its higher chemical stability. The VIP values of the other clusters are less than that of Si_{22} by 0.21 eV–1.00 eV. This suggests that Si_{22} has the higher chemical stability than others, which is in accord with the above analysis based on HOMOLUMO gap. For Si_{20}, Si_{21} and Si_{25} clusters, our calculated VIP values (7.01 eV, 7.07 eV and 6.89 eV) are in agreement with the previous theoretical results (6.76 eV^{51}, 6.85 eV^{44} and 6.49 eV^{44}), respectively.
Polarizability
According to the simple perturbation theory using the oneelectron wave function, the value of polarizability α can be given by the following sumoverstates (SOS) expression^{52,53,54}:
where the one electron matrix elements l and k are the antibonding (or unoccupied) and bonding (or occupied) orbitals, respectively. ε_{l}−ε_{k} is the corresponding HOMOLUMO transition energy. Accurate evaluation of matrix element is not straightforward, because it involves detailed knowledge of the wave functions of orbitals k and l as well as the relative position of each atom. Luckily, when the size of a molecule is large, the matrix element part of the equation can often be considered more or less constant. In this case, the mean polarizability per atom is then given by the invariant trace, where n is the number of atoms in the cluster. As a benchmark test of the method, we have calculated the polarizability of isolated silicon atom. The theoretical value of 3.71 Å^{3}/atom agrees well with the experimental result of 3.70 Å^{3}/atom^{55}. Thus, we can extend the above calculated method to the mediumsized Si_{n} clusters in the range of n = 20–30.
The calculated results for Si_{n}^{μ} (n = 20–30, μ = 0, −1 and +1) clusters are summarized in Table 3, in which the α values from the literature^{55} are also included for comparison. From Table 3, it can be clearly seen that the calculated α values of neutral clusters (n = 20–28) are in good agreement with the previous theoretical results at PBE level, with discrepancy is less than 0.25^{56}. All the α values of clusters are significantly larger than the polarizability of the bulk (3.17 Å^{3}/atom)^{57}. In addition, the polarizabilities of Si_{n} clusters are not size sensitive and close to 4.60 Å^{3}/atom. However, the experimental values of Schäfer et al.^{55} show much larger fluctuations as cluster size with an average value of about 3.5 Å^{3}/atom over the range n = 20–28. This lack of disagreement may be explained by the temperature effects^{58}. Since the average polarizability can only be directly measured in experiments if the static dipole moment of the cluster is zero. For cationic clusters, the calculated results of polarizability also show relatively small variations in the value of a over the size range 20 ≤ n ≤ 30, with all values significantly larger than the bulk limit. In contrast to anionic cluster, the calculated results indicate that the polarizabilities vary strongly and irregularly with size. From Fig. 7, it is interesting to note that there is a clear transition in the value of α occurring at around n = 27. The atomic polarizabilities can be related to the volume occupied by electrons. The compact sphericallike geometries have relatively fewer and shorter bonds, binding the valence electrons tighter with a smaller spatial volume than the prolate structures. Thus, the sphericallike clusters have smaller polarizabilities than the prolate clusters. Once again the above polarizabilities transition have demonstrated that the structure transform from prolate to sphericallike geometries in anionic silicon cluster.
To get a clear insight of the correlation between polarizability and HOMOLUMO gap, the α values and the inverse of HOMOLUMO gaps are plotted as a function of the cluster size n in Fig. 7. As is shown in Fig. 7, the curves of α values is dissimilar to the (HOMOLUMO gap)^{−1} lines. This reveals that there is no such correlation between the polarizability and HOMOLUMO gap among these clusters, which is consistent with the conclusions of Jackson et al.^{59} and Deng et al.^{56}. For example, the neutral Si_{25}, anion Si_{25}^{–} and cation Si_{20}^{+} have the largest polarizability (4.97, 5.64 and 5.32 Å^{3}/atom, respectively) in respective species. If the inverse relationship between polarizability and HOMOLUMO gaps is true, their (HOMOLUMO gap)^{−1} should be the largest values. Namely, these clusters should have the smallest HOMOLUMO gaps as well. However, that is not the case in neutral and anionic Si_{25} cluster (see Table S1). This lost correlation between the polarizability and the HOMOLUMO gap can be attributed to the vanishing matrix element between the HOMO and LUMO in certain clusters of high symmetry. Take neutral Si_{22} cluster as an example, the wave functions of the HOMO and LUMO can be of completely different symmetry (Figure S6). This incompatibility symmetry causes that the HOMOLUMO transition is forbidden. Consequently, all the matrix elements between these occupied and unoccupied orbitals are zero.
Conclusions
The following conclusions emerge from the present combined CALYPSO structure searching method and densityfunctional theory study of mediumsized neutral, anionic and cationic silicon clusters.

i
For each cluster size, an extensive search of the lowestenergy structure has been conducted by considering a number of isomers. The binding energies, HOMOLUMO energy gaps, vertical ionization potentials, adiabatic detachment energies, polarizability including Raman activities and infrared intensities are predicted at the B3PW91/6311 + G* level.

ii
Our structural optimizations indicate that an appreciable structural transition from prolate to sphericallike geometries occur at n = 26 for neutral Si_{n} clusters, n = 27 for anions and n = 25 for cations. This is in agreement with the previous experimental observations and theoretical predictions. In addition, the growth pattern of both neutral and cationic Si_{n} (n = 20–24) clusters shows a similar behavior.

iii
For neutral and cationic Si_{n} (n = 20–30) clusters, the structural stabilities between prolate and sphericallike structures were further verified by binding energies. Moreover, the relative stability analysis is carried out by calculating HOMOLUMO gaps and vertical ionization potential, which shows that Si_{22} cluster has higher stability than the neighboring clusters.

iv
Based on the simple perturbation theory, we have discussed the relationship between the polarizability and HOMOLUMO gaps. The results indicate that the inverse relationship between them does not hold in general by comparing the curves of polarizability and (HOMOLUMO gap)^{−1}. How to explain this phenomenon is still an open topic.
Additional Information
How to cite this article: Ding, L. P. et al. Understanding the structural transformation, stability of mediumsized neutral and charged silicon clusters. Sci. Rep. 5, 15951; doi: 10.1038/srep15951 (2015).
References
Gao, J. F. & Zhao, J. J. Initial geometries, interaction mechanism and high stability of silicene on Ag (111) surface. Sci. Repuk. 2, 861–868 (2012).
Limaye, M. V. et al. Understanding of subband gap absorption of femtosecondlaser sulfur hyperdoped silicon using synchrotronbased techniques. Sci. Repuk. 5, 11466–11478 (2015).
Pittaway, F. et al. Theoretical studies of palladiumgold nanoclusters: PdAu clusters with up to 50 atoms. J. Phys. Chem. C 113, 9141–9152 (2009).
Zhu, X. & Zeng, X. C. Structures and stabilities of small silicon clusters: Ab initio molecularorbital calculations of Si7Si11 . J. Chem. Phys. 118, 3558–3570 (2003).
Yoo, S., Zhao, J., Wang, J. L. & Zeng, X. C. Endohedral silicon fullerenes SiN (27 ≤ N ≤ 39). J. Am. Chem. Soc. 126, 13845–13849 (2004).
Yoo, S. & Zeng, X. C. Motif transition in growth patterns of small to mediumsized silicon clusters. Angew. Chem. Int. Ed. 44, 1491–1495 (2005).
Yoo, S. & Zeng, X. C. Structures and stability of mediumsized silicon clusters. III. reexamination of motif transition in growth pattern from Si15 to Si20 . J. Chem. Phys. 123, 164303 (2005).
Yoo, S. et al. Structures and relative stability of mediumsized silicon clusters. V. lowlying endohedral fullerenelike clusters Si31Si40 and Si45 . J. Chem. Phys. 124, 164311 (2006).
Yoo, S., Shao, N. & Zeng, X. C. Structures and relative stability of medium and largesized silicon clusters. VI. fullerene cage motif for lowlying clusters Si39, Si40, Si50, Si60, Si70 and Si80 . J. Chem. Phys. 128, 104316 (2008).
Arnold, C. C. & Neumark, D. M. Study of Si4 and Si4ˉ using threshold photodetachment (ZEKE) spectroscopy. J. Chem. Phys. 99, 3353–3362 (1993).
Jarrold, M. F. & Constant, V. A. Silicon cluster ions: evidence for a structural transition. Phys. Rev. Lett. 67, 2994–2997 (1991).
Jarrold, M. F. Nanosurface chemistry on sizedselected silicon clusters. Sci. 252, 1085–1092 (1991).
Jarrold, M. F. & Bower, J. E. Mobilities of silicon cluster ions: the reactivity of silicon sausages and spheres. J. Chem. Phys. 96, 9180–9190 (1992).
Shvartsburg, A. A., Hudgins, R. R., Dugourd, P. & Jarrold, M. F. Structural information from ion mobility measurements: applications to semiconductor clusters. Chem. Soc. Rev. 30, 26–35 (2001).
Kaxiras, E. & Jackson, K. Shape of small silicon clusters. Phys. Rev. Lett. 71, 727–730 (1993).
Jarrold, M. F. & Honea, E. C. Dissociation of large silicon clusters: the approach to bulk behavior. J. Phys. Chem. 95, 9181–9185 (1991).
Bachels, T. & Schafer, R. Binding energies of neutral silicon clusters. Chem. Phys. Lett. 324, 365–372 (2000).
Fuke, K., Tsukamoto, K., Misaizu, F. & Sanekata, M. Near Threshold photoionization of silicon clusters in the 248146 nm region: ionization potentials for Sin . J. Chem. Phys. 99, 7808–7812 (1993).
Ma, L. et al. Lowestenergy endohedral fullerene structures of SiN (30 ≤ N ≤ 39) clusters by density functional calculations. Phys. Rev. A 73, 063203 (2006).
Wang, J., Zhao, J., Ma, L. & Wang, G. Firstprinciples study of structural evolution of mediumsized SiN clusters (41 ≤ N ≤ 50) within stuffed fullerene cages. Eur. Phys. J. D 45, 289–294 (2007).
Yoo, S. & Zeng, X. C. Structures and relative stabilities of mediumsized silicon clusters. IV. Motifbased lowlying clusters Si21Si30 . J. Chem. Phys. 124, 054304–054306 (2006).
Wang, Y. C., Lv, J., Zhu, L. & Ma, Y. M. Crystal structure prediction via particleswarm optimization. Phys. Rev. B 82, 094116 (2010).
Wang, Y. C., Lv, J., Zhu, L. & Ma, Y. M. CALYPSO: A method for crystal structure prediction. Comput. Phys. Commun. 183, 2063–2070 (2012).
Wang, Y. C. et al. An effective structure prediction method for layered materials based on 2D particle swarm optimization algorithm. J. Chem. Phys. 137, 224108 (2012).
Lv, J., Wang, Y. C., Zhu, L. & Ma, Y. M. Particleswarm structure prediction on clusters. J. Chem. Phys. 137, 084204 (2012).
Zhu, L. et al. Reactions of xenon with iron and nickel are predicted in the Earth’s inner core. Nature. Chem. 6, 644–648 (2014).
Lu, S. H. et al. Selfassembled ultrathin nanotubes on diamond (100) surface. Nature. Commun. 5, 3666–3672 (2014).
Frisch, M. J. et al. Gaussian 09 (Revision C.0). Gaussian, Inc., Wallingford, CT, 2009.
Cheeseman, J. R., Trucks, G. W., Keith, T. A. & Frisch, M. J. A comparison of models for calculating nuclear magnetic resonance shielding tensors. J. Chem. Phys. 104, 5497–5509 (1996).
Adamo, C., Matteo, A. di & Barone, V. Tuning of structural and magnetic properties of nitronyl nitroxides by the environment. A combined experimental and computational study. Adv. Quantum. Chem. 36, 45–47 (1999).
Becke, A. D. Densityfunctional thermochemistry. III. the role of exact exchange. J. Chem. Phys. 98, 5648–5652 (1993).
Lee, C. T., Yang, W. T. & Parr, R. G. Development of the collesalvetti correlationenergy formula into a functional of the electron density. Phys. Rev. B 37, 785–789 (1988).
Perdew, J. P. & Wang, Y. Accurate and simple analytic representation of the electrogas correlation energy. Phys. Rev. B: Condens. Matter. 45, 13244–13249 (1992).
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).
Perdew, J. P., Ziesche, P. & Eschrig, H. Electronic Structure of Solids, ed. Akademie Verlag, Berlin, 1991.
Barlett, R. J. & Musia, M. Coupledcluster theory in quantum chemistry. Rev. Mod. Phys. 79, 291–352 (2007).
Kitsopoulos, T. N., Chick, C. J., Zhao, Y. & Neumark, D. M. Study of the lowlying electronic states of Si2 and Si2 using negative ion photodetachment techniques. J. Chem. Phys. 95, 1441–1448 (1991).
Huber, K. P. & Herzberg, G. Molecular Spectra and Molecular Structure, Constants of Diatomic Molecules, vol. IV, Van Nostrand Reinhold, New York, 1979.
Dixon, D. A., Feller, D., Peterson, K. A. & Gole, J. L. The molecular structure and ionization potential of Si2: The role of the excited states in the photoionization of Si2 . J. Phys. Chem. A 104, 2326–2332 (2000).
Marijnissen, A. & Ter Meulen, J. J. Determination of the adiabatic ionization potentials of Si2 and SiCl by photoionization efficiency spectroscopy. Chem. Phys. Lett. 263, 803–810 (1996).
Bai, J. et al. Structural evolution of anionic silicon clusters SiN (20 ≤ N ≤ 45). J. Phys. Chem. A 110, 908–912 (2006).
Zhu, X. L., Zeng, X. C., Lei, Y. A. & Pan, B. Structures and stability of medium silicon clusters. II. ab initio molecular orbital calculations of Si12Si20 . J. Chem. Phys. 120, 8985–8965 (2004).
Nigam, S., Majumder, C. & Kulshreshtha, S. K. Structural and electronic properties of Sin, Sin^{+} and AlSin (n = 213) clusters: theoretical investigation based on ab initio molecular orbital theory. J. Chem. Phys. 121, 7756–7763 (2004).
Yoo, S., Zeng, X. C., Zhu, X. & Bai, J. Possible lowestenergy geometry of silicon clusters Si21 and Si25 . J. Am. Chem. Soc. 125, 13318–13319 (2003).
Ho, K. M. et al. Structures of mediumsized silicon clusters. Nature 392, 582–585 (1998).
Hudgins, R. R., Imai, M. & Jarrold, M. F. Highresolution ion mobility measurements for silicon cluster anions and cations. J. Chem. Phys. 111, 7865–7870 (1999).
Rata, I. et al. Singleparent evolution algorithm and the optimization of Si clusters. Phys. Rev. Lett. 85, 546–549 (2000).
Jackson, K. A. et al. Unraveling the shape transformation in silicon clusters. Phys. Rev. Lett. 93, 013401–013404 (2004).
Hoffmann, M. A. et al. Ultraviolet photoelectron spectroscopy of Si4^{−} to Si1000^{−}. Eur. Phys. J. D 16, 9–11 (2001).
Pisani, C. Quantum Mechanical Ab Initio Calculation of the Properties of Crystalline Materials. Springer, Berlin, 1996, p. 183.
Liu, B. et al. Ionization of mediumsized silicon clusters and the geometries of the cations. J. Chem. Phys. 109, 9401–9409 (1998).
Brieger, M., Renn, A., Sodeik, A. & Hese, A. Ionization of mediumsized silicon clusters and the geometries of the cations. J. Chem. Phys. 75, 1–8 (1983).
Brieger, M. The Origin of the infrared multiphoton induced luminescence of chromyl chloride. J. Chem. Phys. 89, 275–295 (1984).
Bégué, D., Mérawa, M. & Pouchan, C. Dynamic dipole and quadrupole polarizabilities for the ground 2^{1}S and the lowlying 3^{1}S and 3^{3}S states of Be. Phys. Rev. A 57, 2470–2473 (1998).
Schäfer, R., Schlecht, S., Woenckhaus, J. & Becker, J. A. Polarizabilities of isolated semiconductor clusters. Phys. Rev. Lett. 76, 471–474 (1996).
Deng, K., Yang, J. & Chan, T. Polarizabilities of isolated semiconductor clusters. Phys. Rev. A 61, 025201 (2000).
Pouchan, C., Bégué, D. & Zhang, D. Y. Calculated polarizabilities of small Si clusters. J. Chem. Phys. 121, 4628–4634 (2004).
Jackson, K., Pederson, M., Wang, C. Z. & Ho, K. M. Calculated polarizabilities of intermediatesize Si clusters. Phys. Rev. A 59, 3685–3687 (1999).
Jackson, K. A., Yang, M., Chaudhuri, I. & Frauenheim, Th. Shape, polarizability and metallicity in silicon clusters. Phys. Rev. A 71, 033205 (2005).
Acknowledgements
This work is supported by the the National Natural Science Foundation of China (Nos. 11274235 and 11304167), the Doctoral Education Fund of Education Ministry of China (Nos. 20100181110086 and 20111223070653), Postdoctoral Science Foundation of China (No. 20110491317) and Open Project of State Key Laboratory of Superhard Materials (No. 201405)
Author information
Authors and Affiliations
Contributions
C.L. conceived the idea. L.P.D., C.L. and J.L. performed the calculations. L.P.D., F.H.Z., Y.S.Z., X.Y.K. and P.S. wrote the manuscript and all authors contributed to revisions.
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Electronic supplementary material
Rights and permissions
This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
About this article
Cite this article
Ding, L., Zhang, F., Zhu, Y. et al. Understanding the structural transformation, stability of mediumsized neutral and charged silicon clusters. Sci Rep 5, 15951 (2015). https://doi.org/10.1038/srep15951
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/srep15951
This article is cited by

Decrypting the Structural, Electronic and Spectroscopic Properties of GeMgn+(n = 2–12) Clusters: A DFT Study
Journal of Cluster Science (2022)

Aluminumdoped silicon nanocage and borondoped carbon nanocage as catalysts to oxygen reduction reaction (ORR): a computational investigation
Ionics (2020)

Theoretical Study on the Growth Behavior and Photoelectron Spectroscopy of LanthanumDoped Silicon Clusters LaSi n 0/− (n = 6–20)
Journal of Cluster Science (2019)

Ironbased magnetic superhalogens with pseudohalogens as ligands: An unbiased structure search
Scientific Reports (2017)

Probing the structural evolution of ruthenium doped germanium clusters: Photoelectron spectroscopy and density functional theory calculations
Scientific Reports (2016)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.