CHEM. RES. CHINESE UNIVERSITIES 2010, 26(6), 996 1001 Fragmentation Behavior and Ionization Potentials of Lead Clusters Pb n (n 30) LI Xiao-ping 1, ZHANG Wei 1, LÜ Wen-cai 1,2*, WANG Cai-zhuang 3 and HO Kai-ming 3 1. State Key Laboratory of Theoretical and Computational Chemistry, Institute of Theoretical Chemistry, Jilin University, Changchun 130021, P. R. China; 2. Laboratory of Fiber Materials and Modern Textile, the Growing Base for State Key Laboratory, College of Physics, Qingdao University, Qingdao 266071, P. R. China; 3. Ames Laboratory of US DOE and Department of Physics and Astronomy, Iowa State University, Ames IA 50011, USA Abstract The properties of Pb n (n=2 30) clusters including binding energies, second differences in energy, and HOMO-LUMO gaps, especially fragmentation energies and ionization potentials, have been studied by ab initio calculation. The main fragmentation products of Pb n are shown to be PbPb n 1 for n 14 and two small cluster fragments for larger ones with n>14. The Pb 13 appears frequently as the products in the fragmentations of large clusters. Also, the calculated ionization potentials of the clusters are consistent with the experiment data. Keywords Pb cluster; Fragmentation; Ionization potential Article ID 1005-9040(2010)-06-996-06 1 Introduction The study of small elemental clusters has been extremely active in the current researches because clusters can bridge the gap between individual atom and bulk structure [1 9]. The group-iv elements are of particular interest because of their importance in fundamental as well as applied sciences. The group-iv elements start with non-metallic elements C, Si, Ge and end with the metallic Pb. The clusters of these elements exhibit different structures, energies and properties. While Si, Ge, and Sn clusters have been studied extensively, Pb clusters has also attracted more attention recently. Experimentally, ion mobility measurement has been shown to be a very useful approach to obtain the structure information of atomic clusters. Shvartsburg et al. [10] used ion mobility measurement to probe the structures of Pb n up to n=32, and found that these structures are compact near-spherical morphologies in the whole range. The vertical detachment energies(vdes) of Pb n [11] have been measured on a simple magnetic bottle time-of-flight photoelectron spectrometer in the size range of n=24 204. Waldschmidt et al. [12] studied the fragmentation of Pb n cluster ions(n=2 20) in low-energy collisions with a highly oriented pyrolytic graphite surface by means of a tandem time-of-flight mass spectrometer. All the clusters in this range lost an atom dominantly, the behavior is characteristic for metallic element clusters. Laihing et al. [13] found that Pb 10 is a magic cluster and the abundance pattern characteristics of atom closepacking geometries are prominent in lead clusters using laser photoionzation mass spectroscopy. Theoretically, Lai et al. [14] have studied the structures of the Pb n clusters with 3<n<56 using the n-body Gupta potential to account for the interactions between the atoms in clusters. Mazzone [15] studied the properties of lead clusters in a size range of 10<n<90 [16,17] using a semi-empirical method. Doye et al. studied the stable structures of lead clusters with up to n 150 using glue potential and Gupta potential, respectively, and found the low-energy structures are very sensitive to the potential used in the calculation. Candidate structures for the Pb n clusters up to n=22 were investigated by Wang et al. [18] via a genetic algorithm(ga) simulation with an empirical potential followed by structure refinement by density function theory(dft) calculations using the BLYP functional in DMol package. They have found that the structures *Corresponding author. E-mail: wencailu@jlu.edu.cn Received March 3, 2010; accepted May 20, 2010. Supported by the National Natural Science Foundation of China(Nos.20773047 and 21043001).
No.6 LI Xiao-ping et al. 997 of small clusters of Pb n (n 10) are similar to those of Si, and Ge clusters, and the competition between the atom-centered close-packed structure and the stacked layered structure appears in a size range of 14 22. Rajesh et al. [19] studied the Pb n (n=2 15) clusters with the DFT method under the generalized gradient approximation(gga), and found that Pb clusters favor compact spherical structures with five-fold or six-fold symmetries. Recently they [20] have also investigated the charged Pb n (n=2 15) clusters and found that the structures have similar motifs as the neutral ones. Li et al. [5,6] reported the structures of Pb n (n=2 20) and Pb n (n=21 30) clusters using the DFT(Density function theory)-paw(projector augmented wave)-pbe (Perdew-Burke-Ernzerhof)(DFT-PAW-PBE) method, respecttively, and found that the structures of the Pb clusters are near-spherical which differ from those of Si and Ge clusters in the same size range. We noted that so far most of computational studies have been focused on searching for the low-energy structures. Much less efforts have been devoted to analyzing the properties of Pb clusters. In order to gain a better understanding of the properties and stability of lead clusters, we have carried out a systematic study for the structures of the Pb clusters in a size range of n=2 30. Based on these structures, binding energies, second differences in energy and HOMO-LUMO gaps have been calculated. In particular, we also carried out a detailed study of the ionization potentials and fragmentation behavior of Pb 2 Pb 30 clusters. These studies can provide useful information for a better understanding of the stability of Pb clusters. 2 Computational Methods The geometry optimizations of Pb clusters were performed via the density function theory(dft) with the projector augmented wave(paw) pseudopotential and plane wave(pw) basis set in the vienna ab-initio simulation package(vasp) code [21 23]. The cutoff energy of PW in the calculations was taken to be 98.0 ev. A simple cubic supercell of a size of 20.0 nm was used. The geometry optimization of each isomer was carried out till the energy was converged to an accuracy of 10 5 ev. Considering the importance of spin-orbit coupling effect [24], we have carried out the total energy calculations of the lowest-energy isomers of Pb clusters with the spin-orbit coupling effect. 3 Results and Discussion 3.1 Pb n (n=2 30) Structures The lowest-energy structures of lead clusters(n=2 30) are presented in Fig.1. One can see that Pb n clusters with n 7 have similar structures as the Si, Ge, and Sn clusters. Pb 8 and Pb 9 are formed by capping one or two atoms on the pentagonal bi-pyramid Pb 7 structure. The lowest-energy structure of Pb 10 is a capped trigonal prism, while Pb 13 favors an encapsulated icosahedron structure, and Pb 15 is an encapsulated hexagonal structure. These structures are consistent with those of the previous studies [19]. Pb 16 is composed of two Pb 6 units linked by a Pb 4 ring, leading to a 1-5-4-5-1 layer-staking, Pb 17 has a C 2v symmetry. It can be considered as a perfect cage. Pb 18 favors a cage structure(c 3v ) with a 1-6-1-6-3-1 stacking, and Pb 19 has a structure with a 1-5-6-1-5-1 stacking. Pb 20 has a structure with a D 2h symmetry, which can be considered to be composed of two Pb 7 units linked by a hexagon. In the size range of n=16 20, the compact near-spherical structures are dominant in lead clusters although some structures with the layer motif do exist. The lowest-energy structure of Pb 21 (D 3h ) is formed by two Pb 12 units, in which the two Pb 12 units share three atoms. Pb 22 is an endohedral cage structure with a pentagonal and a quadrangled unstable rings on the cluster surface. Pb 23 is formed by capping an atom at the bottom of Pb 22. Pb 24 is similar to Pb 23, except that the top ring changes from a pentagon to a hexagon, which has a C 3v symmetry and it has a perfect surface structure. Pb 25 can be seen as two atoms capping on the structure of Pb 23, which makes two seven-membered rings. Pb 26, with a C s symmetry, is formed by adding an atom on the top ring of Pb 25. For the Pb n (n=27 30) clusters, the most stable structures belong to the same pattern. They can be seen as Pb 24 with a fragment attached, and in the center there are three endohedral atoms to maintain the cage structure. 3.2 Relative Stabilities In order to understand the relative stability of the Pb n clusters, we have calculated binding energies, second differences in energy, HOMO-LUMO(highest occupied molecular orbital-lowest unoccupied molecular orbital) gaps. All the results presented in this section were obtained from the lowest-energy isomers
998 CHEM. RES. CHINESE UNIVERSITIES Vol.26 as shown in Fig.1. Fig.1 Lowest-energy structures of neutral Pb 2 30 clusters calculated at the DFT-PBE level The binding energy of per atom of Pb n was calculated as E b (Pb n )=[ne(pb) E(Pb n )]/n as a function of ing that these clusters have large binding energies and clusters with n=4, 7, 10, 13, 15, 17, 24 and 28, show- cluster size. As shown in Fig.2, the binding energy of special stabilities. per atom increases rapidly with the cluster size up to The second difference in energy of Pb n is calculated according to Δ 2 E=[E(Pb n1 )E(Pb n 1 ) 2E(Pb n )]. n=8, oscillates in a size range of 9 19, and then becomes smoother in a range of n=20 30. Small humps Δ 2 E as a function of cluster size is plotted in Fig.3. on the curve indicate higher stability for some specific The curve shows an oscillating behavior, and Pb 4, Pb 7, clusters. In particular, there are peaks for the Pb n Fig.2 Binding energies of per atom of Pb n (n=2 30) calculated at the DFT-PBE level Fig.3 Second differences in energy, defined by 2 E=E n 1 E n1 2E n, of the lowest-energy Pb n (n=2 29) structures calculated at the DFT-PBE level
No.6 LI Xiao-ping et al. 999 Pb 13, Pb 15, Pb 17, Pb 24 and Pb 28 correspond to local maxima, suggesting that these clusters are more stable than their neighbors. The HOMO-LUMO gaps of the clusters are considered to be an important property describing thechemical reactive stability of the clusters. As shown in Fig.4, the HOMO-LUMO gaps of the clusters are large at n=4, 10, 13, 16, 20, 28. From the above analyses, Pb 4, Pb 10, Pb 13, and Pb 28 have both large energetic stability as well as reactive stability. In fact, the binding energy reflects thermodynamic stability and the HOMO-LUMO gap shows potential chemical reactivity, they have no direct relationship. For example, Pb 7 has a large binding energy, however, its HOMO-LUMO gap is small. This indicates that Pb 7 has a better thermodynamic stability, but it is more reactive due to the smaller gap. Fig.4 HOMO-LUMO gaps of the lowest-energy Pb n (n=2 30) clusters calculated at the DFT-PBE level 3.3 Fragmentation Behavior Study fragmentation channels and fragmentation energies can provide useful information for understanding the cluster stabilities. It has been found that some small clusters which frequently appear in the fragmentation products are likely to be relatively stable clusters. In the present work we have analyzed the fragmentation of the neutral and charged clusters. Pb n Pb m Pb n m, Pb n Pb m Pb n m The fragmentation energy can be calculated via E f =E(Pb m )E(Pb n m ) E(Pb n ). In this work, we have calculated all possible fragmentation channels, however, we only discuss the energetically most favorable ones. All the results about fragmentations are shown in Fig.5 and Table 1. For the neutral clusters, the calculated results show that although small clusters Pb n (n 12) favor a Fig.5 Fragmentation energies(e f ) of the lowestenergy fragmentation channels of the neutral and cationic Pb clusters versus cluster size monomer evaporation which is the lowest-energy fragmentation channel, the larger clusters(n 13) prefer to dissociate into two small stable clusters, such as Pb 14 Pb 7 Pb 7, Pb 23 Pb 13 Pb 10, and Pb 28 Pb 13 Pb 15. The fragmentation energies of the Pb n (n 12) clusters are larger than those of the larger clusters. Thus the larger clusters dissociate more easily than the small ones. In the fragmentation products, the Pb 7, Pb 10, Pb 13, and Pb 15 are the favorable daughter clusters for several fragmentation processes, which is consistent with the result from the experiment [12]. The results indicate that the lowest-energy dissociation channels of Pb n (n 14) prefer to lose a Pb atom and produce a Pb n 1 fragment. Pb 16 and Pb 18 have the same fragmentation behavior as the smaller charged clusters Pb n(n 14). For n=15, 17, 19, and 20, the most favorable channels of the dissociation are Pb 15 Pb 2 Pb 13, Pb 17 Pb 4 Pb 13, Pb 19 Pb 6 Pb 13, and Pb 20 Pb 7 Pb 13, indicating that Pb 13 appears as a frequent daughter ion in these cluster fragmentations. In the surface-induced dissociation experiment [12], it has been shown that the best channel of fragmentation for Pb n (n 20) is the loss of one atom except Pb 15. In the same experiment, Pb 15 was found to dissociate into Pb 13 and Pb 2, and Pb 13 was observed clearly as a stable fragment. The experimental channels of the fragmentations for Pb 17,19,20 are the loss of one atom. Taking Pb 17 for an example, the best channel observed in the experiment is Pb 17 Pb 1 Pb 16, and there are other channels, such as Pb 17 Pb 2 Pb 15,, Pb 17 Pb 7 Pb 10, Pb 17 Pb 8 Pb 9. The fragmentation snippets were also observed, especially, Pb 10 appeared as a fragment ion for the first time with the incident ener
1000 CHEM. RES. CHINESE UNIVERSITIES Vol.26 Table 1 Fragmentation channels and fragmentation energies(e f ) of Pb n (n=2 30) and Pb n (n=2 30) clusters Cluster E f /ev Fragmentation E f /ev Fragmentation channel(cationic)(m, n m) * n (Neutral) channel(neutral) (Cationic) PBE GGA/SO [20] Expt. [12] 2 1.3268 (1,1) 2.844 (1,1) (1,1) (1,1) 3 1.8485 (1,2) 2.095 (2,1) (2,1) (2,1) 4 2.1648 (1,3) 2.169 (3,1) (3,1) (3,1) 5 1.521 (1,4) 2.022 (4,1) (4,1) (4,1) 6 2.531 (1,5) 2.499 (5,1) (5,1) (5,1) 7 2.3775 (1,6) 2.545 (6,1) (6,1) (6,1),(5,2) 8 1.3911 (1,7) 1.819 (7,1) (7,1) (7,1),(6,2) 9 2.263 (1,8) 2.220 (8,1) (8,1) (8,1),(7,2) 10 1.9517 (1,9) 1.921 (9,1) (9,1) (9,1),(8,2),(7,3) 11 1.5794 (1,10) 1.906 (10,1) (10,1) (10,1),(9,2) 12 1.9519 (1,11) 1.921 (11,1) (11,1) (11,1),(10,2) 13 1.7777 (7,6) 2.666 (12.1) (12.1) (12.1),(11,2) 14 0.7048 (7,7) 1.022 (13,1) (13,1) (13,1),(12,2) 15 1.7071 (7,8) 2.144 (13,2) (13,2) (13,2),(12,3) 16 1.0665 (7,9) 1.717 (15,1) (15,1),(13,3) 17 1.0696 (7,10) 1.994 (13,4) (16,1),(15,2) (10,7),(9,8) 18 0.7229 (9,9) 1.551 (17,1) (17,1),(16,2) (10,8) 19 0.7641 (10,9) 1.403 (13,6) (18,1),(17,2) (10,9),(9,10) 20 0.6957 (10,10) 0.872 (13,7) (19,1),(18,2) (10,10),(13,7) 21 0.8684 (9,12) 1.330 (13,8) 22 0.8212 (7,15) 1.144 (13,9) 23 0.7436 (10,13) 1.167 (13,10) 24 0.9876 (9,15) 1.560 (17,7) 25 0.5747 (10,15) 1.292 (13,12) 26 0.5867 (13,13) 1.185 (13,13) 27 0.8715 (10,17) 1.381 (17,10) 28 1.0352 (13,15) 1.552 (13,15) 29 0.9456 (13,16) 1.555 (22,7) 30 0.6006 (15,15) 1.274 (23,7) * In the fragmentation channel(cationic), the former part(m) is the cationic Pb m cluster and the latter(n m) is the neutral Pb n m cluster. increased [12]. Experimentally, the incident energy has an important effect on the fragmentations of Pb n clusters. In the experiment of Ref.[12], E i is 0.2 0.4 ev/atom, so Pb 17 PbPb 16 can easily be observed. We expect that when the E i increases the calculated lowest-energy channel, Pb 17 Pb 4 Pb 13 should be observable. It is also known that the fragmentation processes may have dissociation barriers. In our present study, however, we only considered the total energy difference between reactant and product. Therefore, the incident energy and the possible dissociation barriers could be the reasons for the discrepancy between our theoretical prediction and the experimental observation for some Pb cluster fragmentations. It is of interest to note that as a fragmentation daughter ion, Pb 13 appears in the fragmentation products of many larger charged Pb n clusters such as n=21, 22, 23, 25, 26, 28. However, in the other larger charged clusters Pb n (n=24, 29, 30), Pb 7 is found to be the most favorable daughter cluster. Our calculation shows that the fragmentations which produce Pb 7 such as Pb 24 Pb 7 Pb 17, Pb 29 Pb 7 Pb 22 and Pb 30 Pb 7 Pb 23 are the energetically most favorable channels for these clusters such as Pb 24, Pb 29, Pb 30. This is consistent with the experimental observation that Pb 7 is abundant in photoelectron mass spectroscopy [13]. In fact, the multi-step fragmentations for large clusters, such as Pb 24 Pb 7 Pb 17 Pb 7 (Pb 4 Pb 13 ), may also occur. In this work, we only discuss the one-step fragmentations of the lead clusters. From the fragmentation analysis, we can know that the small Pb cluster ions do favor the loss of one atom, consistent with the metal cluster fragmentation behavior. However, as the cluster size increases, two small daughter clusters could be produced in fragmentations. 3.4 Ionization Potentials The ionization potential is an important character for understanding the electronic properties of clusters, and it can also account for the metallicity of the clus-
No.6 LI Xiao-ping et al. 1001 ters. We have calculated the adiabatic ionization potential of the Pb n clusters from the difference in total energy between the neutral and cationic clusters. The comparisons among our calculated result, the experimental data [25], and the result calculated by GGA/SO [20] are shown in Fig.6. It can be seen from the plot that there is a rapid decrease in the ionization potential as the cluster size increases. In particular, there is a sharp drop in the ionization potential of Pb 13. The threshold IP values for n=2 4 are larger by more than 0.5 ev compared to those of other clusters studied in this work. For the larger clusters, the IP curve becomes much smoother. Since when the IP gets smaller, the cluster will be more close to a metallic system, the larger Pb clusters are more metallic compared to small ones. The overall trend in the adiabatic ionization potential obtained from our calculations agrees well with the result of the photoionzation spectroscopy measurement, though the calculated IPs are slightly smaller than the experimental values. Fig.6 Calculated ionization potentials(ips) versus cluster size for Pb n with n=2 30, compared with the GGA/SO calculated results [20] and the photoionization experiment [25] 4 Conclusions We have performed DFT calculations to study the properties of Pb n and Pb n (n=2 30) clusters, including binding energies, second differences in energy, HOMO-LUMO gaps, fragmentation behavior, and ionization potential. The calculated results show that Pb 4, Pb 10, Pb 13 and Pb 28 have both larger energetic stability and larger reactive stability than their neighbors. The calculated fragmentation energies and the predicted dissociation products reveal that the most dominant channels of Pb n clusters are the evaporation of an atom for small clusters with n 14. In the dissociation of larger clusters, the fragmentations can produce two daughter clusters, and Pb 13 appears frequently in the fragmentation products. In addition, the results of the calculated IPs for the Pb n (n=2 30) clusters also agree with the experimental result. References [1] Zhao L. Z., Lü W. C., Qin W., Wang C. Z., Ho K. M., J. Phys. Chem. A, 2008, 112, 5815 [2] Zhang W., Li Z., Zhang G., Lü W. C., Chem. Res. Chinese Universities, 2010, 26(2), 294 [3] Zhao L. Z., Lü W. C., Qin W., Zang Q. J., Wang C. Z., Ho K. M., Chem. Phys. Lett., 2008, 455, 225 [4] Majumder C., Kumar V., Mizuseki H., Kawazoe Y., Phys. Rev. B, 2001, 64, 233405 [5] Li X. P., Lü W. C., Zang Q. J., Chen G. J., Wang C. Z., Ho K. M., J. Phys. Chem. A, 2009, 113, 6217 [6] Li X. P., Lü W. C., Wang C. Z., Ho K. M., J. Phys. Condens. Matter, 2010, 22, 465501 [7] Jackson K. A., Horoi M., Chaudhuri I., Frauenheim T., Shvartshurg A. A., Phys. Rev. Lett., 2004, 93, 013401 [8] Hunter J. M., Fye J. L., Jarrold M. F., Bower J. E., Phys. Rev. Lett., 1994, 73, 2063 [9] Shvartsburg A. A., Jarrold M. F., Phys. Rev. A, 1999, 60, 1235 [10] Shvartsburg A. A., Jarrold M. F., Chem. Phys. Lett., 2000, 317, 615 [11] Luder C., Meiwes-Broer K. H., Chem. Phys. Lett., 1998, 294, 391 [12] Waldschmidt B. B., Turra M., Schafer R., Z. Phys. Chem., 2007, 221, 1569 [13] Laihing K., Wheeler R. G., Wilson W. L., Duncan M. A., J. Chem. Phys., 1987, 87, 3401 [14] Lai S. K., Hsu P. J., Wu K. L., Liu W. K., Iwamatsu M., J. Chem. Phys., 2002, 117, 10715 [15] Mazzone A. M., Phys. Rev. B, 1996, 54, 5970 [16] Doye J. P. K., Hendy S. C., Eur. Phys. J. D., 2003, 22, 99 [17] Doye J. P. K., Computationals Materials Sciences, 2006, 35, 227 [18] Wang B. L., Zhao J. J., Chen X. S., Shi D. N., Wang G. H., Phys. Rev. A, 2005, 71, 033201 [19] Rajesh C., Majumder C., Rajan M. G. R., Kulshreshtha S. K., Phys. Rev. B, 2005, 72, 235411 [20] Rajesh C., Majumder C., J. Chem. Phys., 2007, 126, 244704 [21] Kresse G., Joubest D., Phys. Rev. B, 1999, 59, 1758 [22] Kresse G., Hafner J., Phys. Rev. B, 1993, 47, 558 [23] Kresse G., Furthmuller J., Phys. Rev. B, 1996, 54, 11169 [24] Zhao C. Y., Balasubramanian K., J. Chem. Phys., 2002, 116, 10287 [25] Saito Y., Yamauchi K., Mihama K., Jpn. J. Appl. Phys., 1982, 21, L396