- Email: [email protected]
Size-dependence of stability and optical properties of lead sulfide clusters
Chemical Physics Letters 457 (2008) 163–168
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Size-dependence of stability and optical properties of lead sulfide clusters Jiangang He, Caiping Liu, Fujun Li, Rongjian Sa, Kechen Wu * State Key Laboratory of Structural Chemistry, Fujian Institute of Research on the Structure of Matter, Graduate School of the Chinese Academy of Sciences, Chinese Academy of Sciences, 155 Yangqiao Road W., Fuzhou, Fujian 350002, PR China
a r t i c l e
i n f o
Article history: Received 7 January 2008 In final form 28 March 2008 Available online 7 April 2008
a b s t r a c t The geometries, stabilities, and optical properties of lead sulfide (PbS)n clusters (n = 1–16) have been studied by using density functional theory method in order to exposit the structural evolutions and size-dependent stabilities and optical properties. The clusters favor the galena structures and follow a simple cubic growth pattern. The even number clusters show higher stabilities than the odd ones do and the magic numbers appear at n = 4, 8, 10, and 14. The mean static polarizabilities of thelowest-energy structures exhibit a linearly increased tendency while the largest mean second hyperpolarizabilities of each size present the exponentially increased character. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction In recent years, semiconductor clusters have attracted much attention due to their unique size-dependent electronic and structural properties that are different from both the extended bulk and the atomic states [1–4]. The building blocks of semiconductor clusters have therefore promising applications as various size-tunable optical materials and molecular devices, such as laser materials, solar cell, light-emitting diodes (LED) and so on [5–7]. As an important direct band gap semiconductor material, lead sulfide (PbS) possesses the merits of narrow band gap (0.41 eV), large excitation Bohr radius (18 nm), and high carrier mobility [8,9]. Moreover, the favorable electronic and optical properties of PbS are useful in producing various devices [10,11]. The electronic structures, excitation properties, and optical properties of nano-scaled PbS clusters have been thoroughly studied both experimentally and theoretically. Cademartiri et al. reported the sizedependent transition energies and the extinction coefficients of PbS nanocrystals [12]. Bakueva et al. found the size-tunable infrared electroluminescence from PbS nanocrystals in semiconducting polymers [13]. Kane et al. theoretically investigated the electronic structures of PbS nanocrystals using semi-empirical tight binding method [14] and their results show that the band gap is closely correlated with the crystal sizes. Recently Zeng et al. studied the geometric and electronic structures of small (PbS)n (n = 1–9) clusters at the level of density functional theory (DFT) [15]. They found these small (PbS)n clusters exhibit galena crystal character and the HOMO–LUMO gaps show an obviously oscillating character. Some experimental studies further indicated the nonlinear optical (NLO) activities of PbS nano-scale materials. Kim et al. reported that the * Corresponding author. Fax: +86 591 8379 2932. E-mail address: [email protected] (K. Wu). 0009-2614/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2008.03.085
intra-zeolite PbS quantum dots show very high third-order NLO activities [16]. Li et al. found that PbS nanorods exhibit large nonlinear absorption coefficient and refractive index [17]. Buso et al. studied the PbS nanocrystals doped silica films and found these materials show high optical nonlinearities [18]. However, to the best of our knowledge, few theoretical works focus on size-dependent linear and nonlinear optical properties of interesting PbS clusters. This motivates us to investigate the structural evolutions and size-dependent linear and nonlinear optical properties of PbS clusters up to the medium size within the density functional theory framework, which will be helpful to understand the NLO origin and to explore the relevant NLO material at nano-scale. 2. Computational methods The initial geometries of (PbS)n clusters (n = 1–16) were constructed by energy-surface searching as well as referring to the structures presented in Ref. [15] and some typical binary clusters such as (ZnO)n and (AlN)n. The full optimizations were performed by using the hybrid B3LYP [19,20] exchange-correlation (XC) functional and effective core potential (ECP) basis set SBKJC [21,22] for lead and 6-31+G for sulfur. The similar computational scheme has been confirmed to be suitable for small (PbS)n (n = 1–9) clusters [15]. All the optimized geometries have been verified to be the local energy minima by the harmonic vibrational frequency calculation. The energies of (PbS)n clusters were calculated with the same XC functional but a larger basis set: SBKJC+d (exponent = 0.149) [15] and 6-311+G were used for lead and sulfur, respectively. The excitation properties of clusters were calculated by using the TDDFT method at the same level of theory as the energy calculations. The static (hyper)polarizability were calculated by means of finite-field (FF) approach [23], which has recently been extensively
164
J. He et al. / Chemical Physics Letters 457 (2008) 163–168
used for the sizable molecule and clusters [24–26]. The various electric field intensities from 0.001 a.u. to 0.008 a.u. have been tested and the intensity of 0.002 a.u. was found to be suitable for all the clusters. The mean static polarizability and the second hyperpolarizability are evaluated in terms of the following definition,
1 ðaxx þ ayy þ azz Þ 3 1 hci ¼ ðcxxxx þ cyyyy þ czzzz þ 2cxxyy þ 2cyyzz þ 2czzxx Þ 5
hai ¼
ð1Þ ð2Þ
All the calculations were performed by GAUSSIAN 03 program package [27].
Fig. 1. The lowest-energy and low-lying isomers of (PbS)n (n = 1–16) clusters. The yellow balls represent S atoms and the black ones denote Pb atoms. The symmetry and energy difference with respect to the lowest-energy structure of all isomers are also listed. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
J. He et al. / Chemical Physics Letters 457 (2008) 163–168
Fig. 1 (continued)
165
166
J. He et al. / Chemical Physics Letters 457 (2008) 163–168
Fig. 1 (continued)
3. Results and discussion 3.1. Equilibrium geometries Fig. 1 presents the optimized geometries of (PbS)n clusters (n = 1–16) along with their corresponding symmetries and relative energies. The low-lying isomers of each size are labeled as a, b, c, etc. in energy order from low to high. It should be noted that only the low-lying isomers with the total energies not more than 1.00 eV higher than that of the lowest-energy structure are presented for simplicity and clarity. The optimized bond length of monomer (n = 1) is 2.282 Å, which agrees well with experimental value (2.287 Å) [28] but it is much smaller than the Pb–S distance (2.957 Å) in lead sulfide crystal [29]. (PbS)2 cluster favors distorted rhombus configuration (C2v) with Pb–S bond lengths of 2.536 Å which is much larger than that of monomer. When n = 3, the cluster exhibits a folded rectangle structure with mean bond length of 2.563 Å. The (PbS)4 cluster shows a cubic structure (Td) with Pb–S bond lengths of 2.669 Å. Cluster 5a, the lowest-energy structure of (PbS)5 clusters, favors double deck ring structure with a tetragonal at the bottom and a distorted hexagon at the top. Cluster 6a is a quadrangular prism structure (D2h) yielded by adding either one monomer to 5c or one dimer to the (PbS)4. The isomer 6c is tubular structure (S6) consisting of two hexagonal rings. The similar configuration was also found for ZnO cluster [30], where the tube had also been confirmed to be metastable. The cage (7a, C3v) made up of six rhombuses and three hexagons is thelowest-energy structure when n = 7. The similar cage-like configuration had also been found in ZnO [30], AlN [31,32], and GaN [33] clusters, but only for AlN cluster it is the most stable configuration [31,32]. We note that the total energy of cage 7a is 1.09 eV lower than that of the lowest-energy configuration of n = 7 in Ref. [15]. Cluster 8a has a typical quadrangular prism structure with D2d symmetry. Cluster 9a has quadrel-like structure (C4v) formed by four cubes. The mean bond length of the cube is 2.764 Å which is longer than that of (PbS)4 and is much closer to Pb–S distance in PbS crystal. The cage (9e, C3h) with six rhombuses and five hexagons is similar to the most stable configurations of (ZnO)9 and (GaN)9 clusters [30,33]. Cluster 10a has a quadrangular prism structure (D2h) and the cage (10f, C3) appeared in ZnO [30,34] and AlN [31,32] cluster is 0.75 eV higher in energy than 10a. Cluster 11a is yielded from 10b or 10c by attaching one
monomer, while the cage 11e (Cs) was reported to be the most stable configuration for GaN cluster [33]. Cluster 12a is again a quadrangular prism structure (D2d). The quadrel-like structure (12c, C2h) coupled by six cubes is similar to 9a and its mean bond length (2.825 Å) is much closer to that of PbS crystal. Cluster 13a can be obtained by adding one monomer to 12a. It is 0.17 and 0.54 eV more stable than the defective quadrel-like (13b, C1) and cage (13c, C3v) clusters, respectively. Cluster 14a has a quadrangular prism structure (D2h) too. Cluster 15a is a distorted quadrel-like structure but it is 0.29 and 0.45 eV more stable than the perfect quadrel-like (15b, C2v) and tubular (15c, D3h) clusters, respectively. Cluster 15c is the longest tube among all the clusters in the present study and its Pb–S mean distance is the longest (2.733 Å) as well. As described above, the quadrangular prism structures are the lowest-energy configurations from n = 4 to 14 when n is even. But quadrel-like 16a is an exception, whose energy is 0.38 eV lower than that of the quadrangular prism cluster 16b. 3.2. Size-dependence of stability The average binding energy (Eb) of cluster (PbS)n is defined as follow: Eb ¼ ½En ðPbSÞ nEðPbÞ nEðSÞ=n
ð3Þ
The effect of basis set superposition errors (BSSE) has been evaluated by the counterpoise calculations [35] for n = 1–6 and is found much smaller than Eb. So we do not consider BSSE effect in the following calculations but the zero-point energy corrections have been taken into account. Fig. 2 plots the evolution of Eb of the lowest-energy structure clusters with respect to the cluster size. The curve increases monotonically when n < 5 and then shows an odd–even oscillation when n increases. In order to show the relative stabilities, we further calculate the second-order difference of energy (D2E) in term of the following formula, D2 E ¼ ½2En ðPbSÞ Enþ1 ðPbSÞ En1 ðPbSÞ
ð4Þ
As shown in Fig. 2, D2E of the clusters with the even n are at peaks, especially for n = 4, 8, 10, and 14. According to the jellium model [36], the particularly high stabilities of clusters can be understood in view of the magic numbers of total valence electrons of 2, 8, 18, 20, 34, 40, 58, etc. The magic numbers of (PbS)n clusters
J. He et al. / Chemical Physics Letters 457 (2008) 163–168
Fig. 2. Size-dependence of the average binding energy and second-order difference of energy for the (PbS)n (n = 1–16) clusters with the lowest-energy structures.
are inconsistent with this model except n = 4 where the valence electron is 40. This indicates that the stability mechanism of (PbS)n clusters is different from alkali metal clusters. 3.3. Static polarizability and second hyperpolarizability The evolution of mean polarizability h a i with cluster size is depicted in Fig. 3a. The h a i of the lowest-energy clusters linearly increase with n, which has also been observed in small GaAs clusters [37]. The clusters with quadrangular prism structures (6a, 8a, 10a, 12a, 14a, and 16b) possess the largest h a i among their isomers in relevant sizes. As shown in Fig. 3b, the curve of h a i/ n with respect to cluster size slowly rises and oscillates with odd–even size. It is strikingly different from the monotonically decreasing behavior of small GaAs [37] and CdSe [38] clusters. The largest mean second hyperpolarizabilities (hci) of each size exponentially increase with n as illustrated in Fig. 4a. And the hci of the lowest-energy clusters (na) increase linearly before n = 6, oscillate from n = 7 to 11, increase linearly again in the region of n = 12–14, drop down drastically at n = 15, and then climb up at n = 16 (see Fig. 4 b). The quadrangular prism structures present the largest h c i among the isomers in relevant sizes. In contrast to h a i, h c i is more geometry sensitive, especially when the cluster size are large. Moreover, the diagonal components of c for quadran-
Fig. 3. Size-dependence of the mean static polarizabilities (a) and of the mean static polarizabilities per unit (b) for the lowest-energy structure of each size.
167
Fig. 4. Size-dependence of the largest mean static second hyperpolarizabilities of each size (a) and of the mean static second hyperpolarizabilities per unit for the lowest-energy structure (b).
gular prism structures are distinctly anisotropic (czzzz cxxxx, c yyyy), especially for large size. In order to elucidate the origin of large h c i of quadrangular prism structures, the three-state model [39] based on the sumover-state (SOS) approach is used, ! 1 M 2rm M 2mv M 4rm M 2rm Dl2rm ciiii ¼ þ ð5Þ 6 E2rm Emv E3rm E3rm where Dlrm is the difference between the dipole moments of the states |r i and |m i, Erm is the transition energy of the states |r i and |m i, and Mrm is the transition dipole moment between the states |r i and |m i. According to this model, the excitation of lower energy transition with larger transition dipole moment will lead to larger c. As mentioned above, the axial czzzz is significantly larger than other tensor components for the quadrangular prism structures. Therefore, czzzz makes major contribution to h ci and the c-related charge transfer (CT) is expected to mainly along the z direction. This is confirmed by the calculated excited state properties of clusters using the TDDFT method. For example, the lowest-energy excitation (3.23 eV) of cluster 12a mainly involves two transitions: from HOMO1 to LUMO+5 and from HOMO to LUMO+6. As illustrated in Fig. 5, the CT processes involved in this excitation mainly occur along the z-axis, which contribute to the large transition
Fig. 5. Contour graphs of the frontier orbital of (PbS)12 a cluster. The yellow balls represent S atoms and the black ones denote Pb atoms. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
168
J. He et al. / Chemical Physics Letters 457 (2008) 163–168
dipole moment and lead to a large czzzz. This result implies that a large transition dipole moment can be obtained through enhancing CT along z direction. For the clusters studied in this paper, it can be achieved by extending the length of clusters at the z direction, which has been confirmed by 14a and 16b clusters. It is important to note that the computations and discussions on the second hyperpolarizability in this study are based on the B3LYP density functional. It has been reported the unreliable predictions of the axial hyperpolarizabilities of organic long-chain polymers by using the conventional DFT functionals [40,41]. The result of the c values of the (PbS)16 clusters with long quadrangular prism structure needs further test, though there might be different between inorganic and organic systems. 4. Conclusion In the present work, we find that the (PbS)n (n = 1–16) clusters favor PbS bulk crystal (galena) structure and follow a simple cubic growth pattern. The turnover point from quadrangular prism to quadrel-like structures occurs at n = 16. The even number size clusters are more stable than the odd ones and the magic numbers appear at n = 4, 8, 10, and 14. In addition, the polarizabilities of the clusters with the lowest-energy structures linearly increase and the largest second hyperpolarizabilities of each size grow exponentially. The large polarizabilities and the second hyperpolarizabilities of quadrangular prism structure originate mainly from the considerable charge transfer along their axes (z) direction. Acknowledgment We acknowledge the financial support of National Natural Science Foundation of China with the projects of 20573114. It is also supported by the MOST projects of 2006DFA43020 and ‘973’ project of 2007CB815307. We thank the Supercomputing Center of CNIC for computer resources. References [1] K.M. Ho et al., Nature 392 (1998) 582. [2] A.D. Yoffe, Adv. Phys 42 (1993) 173. [3] A. Puzder, A.J. Williamson, J.C. Grossman, G. Galli, J. Am. Chem. Soc. 125 (2003) 2786.
[4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41]
B. Goswami, C. Ghosh, S. Pal, P. Sarkar, Int. J. Quantum Chem. 106 (2006) 887. V.I. Klimov, et al. , Science 290 (2000) 314. W.U. Huynh, J.J. Dittmer, A.P. Alivisatos, Science 295 (2002) 2425. D.L. Huffaker, C.C. Lin, J. Shin, D.G. Deppe, Appl. Phys. Lett. 66 (1995) 3096. J.L. Machol, F.W. Wise, R.C. Patel, D.B. Tanner, Phys. Rev. B 48 (1993) 2819. E.A. Albanesi, E.L. Peltzer y Blanca, A.G. Petukhov, Comput. Mater. Sci. 32 (2005) 85. Y.H. Ni, X.W. Wei, J.M. Hong, X. Ma, Cryst. Res. Technol. 41 (2006) 885. H. Kanazawa, S. Adachi, J. Appl. Phys. 83 (1998) 5997. L. Cademartiri, E. Montanari, G. Calestani, A. Migliori, A. Guagliardi, G.A. Ozin, J. Am. Chem. Soc. 128 (2006) 10337. L. Bakueva, S. Musikhin, M.A. Hines, T.-W.F. Chang, M. Tzolov, G.D. Scholes, E.H. Sargent, Appl. Phys. Lett. 82 (2003) 2895. R.S. Kane, R.E. Cohen, R. Silbey, J. Phys. Chem. 100 (1996) 7928. H.X. Zeng, Z.A. Schelly, K. Ueno-Noto, D.S. Marynick, J. Phys. Chem. A 109 (2005) 1616. H.S. Kim, M.H. Lee, N.C. Jeong, S.M. Lee, B.K. Rhee, K.B. Yoon, J. Am. Chem. Soc. 128 (2006) 15070. C. Li, G. Shi, H.Y. Xu, S.Y. Guang, R.H. Yin, Y.L. Song, Mater. Lett. 61 (2007) 1809. D. Buso et al., Chem. Mater. 17 (2005) 4965. A.D. Becke, J. Chem. Phys. 98 (1993) 5648. C. Lee, W. Yang, R.G. Parr, Phys. Rev. B 37 (1988) 785. W.J. Stevens, H. Basch, M. Krauss, J. Chem. Phys. 81 (1984) 6026. W.J. Stevens, M. Krauss, H. Basch, P.G. Jasien, Can. J. Chem. 70 (1992) 612. H.D. Cohen, C.C.J. Roothaan, J. Chem. Phys. 43 (1965) S34. Y. Yang, F.H. Wang, Y.S. Zhou, L.F. Yuan, J.L. Yang, Phys. Rev. A 71 (2005) 013202. J.L. Wang, M.L. Yang, J. Jellinek, G.H. Wang, Phys. Rev. A 74 (2006) 023202. M. Yang, P. Senet, C. Van Alsenoy, Int. J. Quantum Chem. 101 (2005) 535. M.J. Frisch et al., Gaussian, Inc., Wallingford, CT, 2004, GAUSSIAN 03. R.A. Teichman III, E.R. Nixon, J. Mol. Spectrosc. 54 (1975) 78. Y. Noda, K. Masumoto, S. Ohba, Y. Saito, K. Toriumi, Y. Iwata, I. Shibuya, Acta Crystallogr. Sect. C Cryst. Struct. Commun. 43 (1987) 1443. B.L. Wang, S. Nagase, J.J. Zhao, G.H. Wang, J. Phys. Chem. C 111 (2007) 4956. H.-S. Wu, F.-Q. Zhang, X.-H. Xu, C.-J. Zhang, H.J. Jiao, J. Phys. Chem. A 107 (2003) 204. A. Costales, M.A. Blanco, E. Francisco, A.M. Pendás, R. Pandey, J. Phys. Chem. B 110 (2006) 4092. J.J. Zhao, B.L. Wang, X.L. Zhou, X.S. Chen, W. Lu, Chem. Phys. Lett. 422 (2006) 170. J.M. Matxain, J.M. Mercero, J.E. Fowler, J.M. Ugalde, J. Am. Chem. Soc. 125 (2003) 9494. F.B. van Duijneveldt, J.G.C.M. Van Duijneveldt-van de Rijdt, J.H. van Lenthe, Chem. Rev. 94 (1994) 1873. W.D. Knight, K. Clemenger, W.A. de Heer, W.A. Saunders, M.Y. Chou, M.L. Cohen, Phys. Rev. Lett. 52 (1984) 2141. P. Karamanis, D. Bégué, C. Pouchan, J. Chem. Phys. 127 (2007) 094706. P. Karamanis, G. Maroulis, C. Pouchan, J. Chem. Phys. 124 (2006) 071101. J.L. Brédas, C. Adant, P. Tackx, A. Persoons, B.M. Pierce, Chem. Rev. 94 (1994) 243. B. Champagne, et al. , J. Chem. Phys. 109 (1998) 10489. B. Champagne, et al. , J. Phys. Chem. A 104 (2000) 4755.
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Size-dependence of stability and optical properties of lead sulfide clusters Jiangang He, Caiping Liu, Fujun Li, Rongjian Sa, Kechen Wu * State Key Laboratory of Structural Chemistry, Fujian Institute of Research on the Structure of Matter, Graduate School of the Chinese Academy of Sciences, Chinese Academy of Sciences, 155 Yangqiao Road W., Fuzhou, Fujian 350002, PR China
a r t i c l e
i n f o
Article history: Received 7 January 2008 In final form 28 March 2008 Available online 7 April 2008
a b s t r a c t The geometries, stabilities, and optical properties of lead sulfide (PbS)n clusters (n = 1–16) have been studied by using density functional theory method in order to exposit the structural evolutions and size-dependent stabilities and optical properties. The clusters favor the galena structures and follow a simple cubic growth pattern. The even number clusters show higher stabilities than the odd ones do and the magic numbers appear at n = 4, 8, 10, and 14. The mean static polarizabilities of thelowest-energy structures exhibit a linearly increased tendency while the largest mean second hyperpolarizabilities of each size present the exponentially increased character. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction In recent years, semiconductor clusters have attracted much attention due to their unique size-dependent electronic and structural properties that are different from both the extended bulk and the atomic states [1–4]. The building blocks of semiconductor clusters have therefore promising applications as various size-tunable optical materials and molecular devices, such as laser materials, solar cell, light-emitting diodes (LED) and so on [5–7]. As an important direct band gap semiconductor material, lead sulfide (PbS) possesses the merits of narrow band gap (0.41 eV), large excitation Bohr radius (18 nm), and high carrier mobility [8,9]. Moreover, the favorable electronic and optical properties of PbS are useful in producing various devices [10,11]. The electronic structures, excitation properties, and optical properties of nano-scaled PbS clusters have been thoroughly studied both experimentally and theoretically. Cademartiri et al. reported the sizedependent transition energies and the extinction coefficients of PbS nanocrystals [12]. Bakueva et al. found the size-tunable infrared electroluminescence from PbS nanocrystals in semiconducting polymers [13]. Kane et al. theoretically investigated the electronic structures of PbS nanocrystals using semi-empirical tight binding method [14] and their results show that the band gap is closely correlated with the crystal sizes. Recently Zeng et al. studied the geometric and electronic structures of small (PbS)n (n = 1–9) clusters at the level of density functional theory (DFT) [15]. They found these small (PbS)n clusters exhibit galena crystal character and the HOMO–LUMO gaps show an obviously oscillating character. Some experimental studies further indicated the nonlinear optical (NLO) activities of PbS nano-scale materials. Kim et al. reported that the * Corresponding author. Fax: +86 591 8379 2932. E-mail address: [email protected] (K. Wu). 0009-2614/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2008.03.085
intra-zeolite PbS quantum dots show very high third-order NLO activities [16]. Li et al. found that PbS nanorods exhibit large nonlinear absorption coefficient and refractive index [17]. Buso et al. studied the PbS nanocrystals doped silica films and found these materials show high optical nonlinearities [18]. However, to the best of our knowledge, few theoretical works focus on size-dependent linear and nonlinear optical properties of interesting PbS clusters. This motivates us to investigate the structural evolutions and size-dependent linear and nonlinear optical properties of PbS clusters up to the medium size within the density functional theory framework, which will be helpful to understand the NLO origin and to explore the relevant NLO material at nano-scale. 2. Computational methods The initial geometries of (PbS)n clusters (n = 1–16) were constructed by energy-surface searching as well as referring to the structures presented in Ref. [15] and some typical binary clusters such as (ZnO)n and (AlN)n. The full optimizations were performed by using the hybrid B3LYP [19,20] exchange-correlation (XC) functional and effective core potential (ECP) basis set SBKJC [21,22] for lead and 6-31+G for sulfur. The similar computational scheme has been confirmed to be suitable for small (PbS)n (n = 1–9) clusters [15]. All the optimized geometries have been verified to be the local energy minima by the harmonic vibrational frequency calculation. The energies of (PbS)n clusters were calculated with the same XC functional but a larger basis set: SBKJC+d (exponent = 0.149) [15] and 6-311+G were used for lead and sulfur, respectively. The excitation properties of clusters were calculated by using the TDDFT method at the same level of theory as the energy calculations. The static (hyper)polarizability were calculated by means of finite-field (FF) approach [23], which has recently been extensively
164
J. He et al. / Chemical Physics Letters 457 (2008) 163–168
used for the sizable molecule and clusters [24–26]. The various electric field intensities from 0.001 a.u. to 0.008 a.u. have been tested and the intensity of 0.002 a.u. was found to be suitable for all the clusters. The mean static polarizability and the second hyperpolarizability are evaluated in terms of the following definition,
1 ðaxx þ ayy þ azz Þ 3 1 hci ¼ ðcxxxx þ cyyyy þ czzzz þ 2cxxyy þ 2cyyzz þ 2czzxx Þ 5
hai ¼
ð1Þ ð2Þ
All the calculations were performed by GAUSSIAN 03 program package [27].
Fig. 1. The lowest-energy and low-lying isomers of (PbS)n (n = 1–16) clusters. The yellow balls represent S atoms and the black ones denote Pb atoms. The symmetry and energy difference with respect to the lowest-energy structure of all isomers are also listed. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
J. He et al. / Chemical Physics Letters 457 (2008) 163–168
Fig. 1 (continued)
165
166
J. He et al. / Chemical Physics Letters 457 (2008) 163–168
Fig. 1 (continued)
3. Results and discussion 3.1. Equilibrium geometries Fig. 1 presents the optimized geometries of (PbS)n clusters (n = 1–16) along with their corresponding symmetries and relative energies. The low-lying isomers of each size are labeled as a, b, c, etc. in energy order from low to high. It should be noted that only the low-lying isomers with the total energies not more than 1.00 eV higher than that of the lowest-energy structure are presented for simplicity and clarity. The optimized bond length of monomer (n = 1) is 2.282 Å, which agrees well with experimental value (2.287 Å) [28] but it is much smaller than the Pb–S distance (2.957 Å) in lead sulfide crystal [29]. (PbS)2 cluster favors distorted rhombus configuration (C2v) with Pb–S bond lengths of 2.536 Å which is much larger than that of monomer. When n = 3, the cluster exhibits a folded rectangle structure with mean bond length of 2.563 Å. The (PbS)4 cluster shows a cubic structure (Td) with Pb–S bond lengths of 2.669 Å. Cluster 5a, the lowest-energy structure of (PbS)5 clusters, favors double deck ring structure with a tetragonal at the bottom and a distorted hexagon at the top. Cluster 6a is a quadrangular prism structure (D2h) yielded by adding either one monomer to 5c or one dimer to the (PbS)4. The isomer 6c is tubular structure (S6) consisting of two hexagonal rings. The similar configuration was also found for ZnO cluster [30], where the tube had also been confirmed to be metastable. The cage (7a, C3v) made up of six rhombuses and three hexagons is thelowest-energy structure when n = 7. The similar cage-like configuration had also been found in ZnO [30], AlN [31,32], and GaN [33] clusters, but only for AlN cluster it is the most stable configuration [31,32]. We note that the total energy of cage 7a is 1.09 eV lower than that of the lowest-energy configuration of n = 7 in Ref. [15]. Cluster 8a has a typical quadrangular prism structure with D2d symmetry. Cluster 9a has quadrel-like structure (C4v) formed by four cubes. The mean bond length of the cube is 2.764 Å which is longer than that of (PbS)4 and is much closer to Pb–S distance in PbS crystal. The cage (9e, C3h) with six rhombuses and five hexagons is similar to the most stable configurations of (ZnO)9 and (GaN)9 clusters [30,33]. Cluster 10a has a quadrangular prism structure (D2h) and the cage (10f, C3) appeared in ZnO [30,34] and AlN [31,32] cluster is 0.75 eV higher in energy than 10a. Cluster 11a is yielded from 10b or 10c by attaching one
monomer, while the cage 11e (Cs) was reported to be the most stable configuration for GaN cluster [33]. Cluster 12a is again a quadrangular prism structure (D2d). The quadrel-like structure (12c, C2h) coupled by six cubes is similar to 9a and its mean bond length (2.825 Å) is much closer to that of PbS crystal. Cluster 13a can be obtained by adding one monomer to 12a. It is 0.17 and 0.54 eV more stable than the defective quadrel-like (13b, C1) and cage (13c, C3v) clusters, respectively. Cluster 14a has a quadrangular prism structure (D2h) too. Cluster 15a is a distorted quadrel-like structure but it is 0.29 and 0.45 eV more stable than the perfect quadrel-like (15b, C2v) and tubular (15c, D3h) clusters, respectively. Cluster 15c is the longest tube among all the clusters in the present study and its Pb–S mean distance is the longest (2.733 Å) as well. As described above, the quadrangular prism structures are the lowest-energy configurations from n = 4 to 14 when n is even. But quadrel-like 16a is an exception, whose energy is 0.38 eV lower than that of the quadrangular prism cluster 16b. 3.2. Size-dependence of stability The average binding energy (Eb) of cluster (PbS)n is defined as follow: Eb ¼ ½En ðPbSÞ nEðPbÞ nEðSÞ=n
ð3Þ
The effect of basis set superposition errors (BSSE) has been evaluated by the counterpoise calculations [35] for n = 1–6 and is found much smaller than Eb. So we do not consider BSSE effect in the following calculations but the zero-point energy corrections have been taken into account. Fig. 2 plots the evolution of Eb of the lowest-energy structure clusters with respect to the cluster size. The curve increases monotonically when n < 5 and then shows an odd–even oscillation when n increases. In order to show the relative stabilities, we further calculate the second-order difference of energy (D2E) in term of the following formula, D2 E ¼ ½2En ðPbSÞ Enþ1 ðPbSÞ En1 ðPbSÞ
ð4Þ
As shown in Fig. 2, D2E of the clusters with the even n are at peaks, especially for n = 4, 8, 10, and 14. According to the jellium model [36], the particularly high stabilities of clusters can be understood in view of the magic numbers of total valence electrons of 2, 8, 18, 20, 34, 40, 58, etc. The magic numbers of (PbS)n clusters
J. He et al. / Chemical Physics Letters 457 (2008) 163–168
Fig. 2. Size-dependence of the average binding energy and second-order difference of energy for the (PbS)n (n = 1–16) clusters with the lowest-energy structures.
are inconsistent with this model except n = 4 where the valence electron is 40. This indicates that the stability mechanism of (PbS)n clusters is different from alkali metal clusters. 3.3. Static polarizability and second hyperpolarizability The evolution of mean polarizability h a i with cluster size is depicted in Fig. 3a. The h a i of the lowest-energy clusters linearly increase with n, which has also been observed in small GaAs clusters [37]. The clusters with quadrangular prism structures (6a, 8a, 10a, 12a, 14a, and 16b) possess the largest h a i among their isomers in relevant sizes. As shown in Fig. 3b, the curve of h a i/ n with respect to cluster size slowly rises and oscillates with odd–even size. It is strikingly different from the monotonically decreasing behavior of small GaAs [37] and CdSe [38] clusters. The largest mean second hyperpolarizabilities (hci) of each size exponentially increase with n as illustrated in Fig. 4a. And the hci of the lowest-energy clusters (na) increase linearly before n = 6, oscillate from n = 7 to 11, increase linearly again in the region of n = 12–14, drop down drastically at n = 15, and then climb up at n = 16 (see Fig. 4 b). The quadrangular prism structures present the largest h c i among the isomers in relevant sizes. In contrast to h a i, h c i is more geometry sensitive, especially when the cluster size are large. Moreover, the diagonal components of c for quadran-
Fig. 3. Size-dependence of the mean static polarizabilities (a) and of the mean static polarizabilities per unit (b) for the lowest-energy structure of each size.
167
Fig. 4. Size-dependence of the largest mean static second hyperpolarizabilities of each size (a) and of the mean static second hyperpolarizabilities per unit for the lowest-energy structure (b).
gular prism structures are distinctly anisotropic (czzzz cxxxx, c yyyy), especially for large size. In order to elucidate the origin of large h c i of quadrangular prism structures, the three-state model [39] based on the sumover-state (SOS) approach is used, ! 1 M 2rm M 2mv M 4rm M 2rm Dl2rm ciiii ¼ þ ð5Þ 6 E2rm Emv E3rm E3rm where Dlrm is the difference between the dipole moments of the states |r i and |m i, Erm is the transition energy of the states |r i and |m i, and Mrm is the transition dipole moment between the states |r i and |m i. According to this model, the excitation of lower energy transition with larger transition dipole moment will lead to larger c. As mentioned above, the axial czzzz is significantly larger than other tensor components for the quadrangular prism structures. Therefore, czzzz makes major contribution to h ci and the c-related charge transfer (CT) is expected to mainly along the z direction. This is confirmed by the calculated excited state properties of clusters using the TDDFT method. For example, the lowest-energy excitation (3.23 eV) of cluster 12a mainly involves two transitions: from HOMO1 to LUMO+5 and from HOMO to LUMO+6. As illustrated in Fig. 5, the CT processes involved in this excitation mainly occur along the z-axis, which contribute to the large transition
Fig. 5. Contour graphs of the frontier orbital of (PbS)12 a cluster. The yellow balls represent S atoms and the black ones denote Pb atoms. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
168
J. He et al. / Chemical Physics Letters 457 (2008) 163–168
dipole moment and lead to a large czzzz. This result implies that a large transition dipole moment can be obtained through enhancing CT along z direction. For the clusters studied in this paper, it can be achieved by extending the length of clusters at the z direction, which has been confirmed by 14a and 16b clusters. It is important to note that the computations and discussions on the second hyperpolarizability in this study are based on the B3LYP density functional. It has been reported the unreliable predictions of the axial hyperpolarizabilities of organic long-chain polymers by using the conventional DFT functionals [40,41]. The result of the c values of the (PbS)16 clusters with long quadrangular prism structure needs further test, though there might be different between inorganic and organic systems. 4. Conclusion In the present work, we find that the (PbS)n (n = 1–16) clusters favor PbS bulk crystal (galena) structure and follow a simple cubic growth pattern. The turnover point from quadrangular prism to quadrel-like structures occurs at n = 16. The even number size clusters are more stable than the odd ones and the magic numbers appear at n = 4, 8, 10, and 14. In addition, the polarizabilities of the clusters with the lowest-energy structures linearly increase and the largest second hyperpolarizabilities of each size grow exponentially. The large polarizabilities and the second hyperpolarizabilities of quadrangular prism structure originate mainly from the considerable charge transfer along their axes (z) direction. Acknowledgment We acknowledge the financial support of National Natural Science Foundation of China with the projects of 20573114. It is also supported by the MOST projects of 2006DFA43020 and ‘973’ project of 2007CB815307. We thank the Supercomputing Center of CNIC for computer resources. References [1] K.M. Ho et al., Nature 392 (1998) 582. [2] A.D. Yoffe, Adv. Phys 42 (1993) 173. [3] A. Puzder, A.J. Williamson, J.C. Grossman, G. Galli, J. Am. Chem. Soc. 125 (2003) 2786.
[4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41]
B. Goswami, C. Ghosh, S. Pal, P. Sarkar, Int. J. Quantum Chem. 106 (2006) 887. V.I. Klimov, et al. , Science 290 (2000) 314. W.U. Huynh, J.J. Dittmer, A.P. Alivisatos, Science 295 (2002) 2425. D.L. Huffaker, C.C. Lin, J. Shin, D.G. Deppe, Appl. Phys. Lett. 66 (1995) 3096. J.L. Machol, F.W. Wise, R.C. Patel, D.B. Tanner, Phys. Rev. B 48 (1993) 2819. E.A. Albanesi, E.L. Peltzer y Blanca, A.G. Petukhov, Comput. Mater. Sci. 32 (2005) 85. Y.H. Ni, X.W. Wei, J.M. Hong, X. Ma, Cryst. Res. Technol. 41 (2006) 885. H. Kanazawa, S. Adachi, J. Appl. Phys. 83 (1998) 5997. L. Cademartiri, E. Montanari, G. Calestani, A. Migliori, A. Guagliardi, G.A. Ozin, J. Am. Chem. Soc. 128 (2006) 10337. L. Bakueva, S. Musikhin, M.A. Hines, T.-W.F. Chang, M. Tzolov, G.D. Scholes, E.H. Sargent, Appl. Phys. Lett. 82 (2003) 2895. R.S. Kane, R.E. Cohen, R. Silbey, J. Phys. Chem. 100 (1996) 7928. H.X. Zeng, Z.A. Schelly, K. Ueno-Noto, D.S. Marynick, J. Phys. Chem. A 109 (2005) 1616. H.S. Kim, M.H. Lee, N.C. Jeong, S.M. Lee, B.K. Rhee, K.B. Yoon, J. Am. Chem. Soc. 128 (2006) 15070. C. Li, G. Shi, H.Y. Xu, S.Y. Guang, R.H. Yin, Y.L. Song, Mater. Lett. 61 (2007) 1809. D. Buso et al., Chem. Mater. 17 (2005) 4965. A.D. Becke, J. Chem. Phys. 98 (1993) 5648. C. Lee, W. Yang, R.G. Parr, Phys. Rev. B 37 (1988) 785. W.J. Stevens, H. Basch, M. Krauss, J. Chem. Phys. 81 (1984) 6026. W.J. Stevens, M. Krauss, H. Basch, P.G. Jasien, Can. J. Chem. 70 (1992) 612. H.D. Cohen, C.C.J. Roothaan, J. Chem. Phys. 43 (1965) S34. Y. Yang, F.H. Wang, Y.S. Zhou, L.F. Yuan, J.L. Yang, Phys. Rev. A 71 (2005) 013202. J.L. Wang, M.L. Yang, J. Jellinek, G.H. Wang, Phys. Rev. A 74 (2006) 023202. M. Yang, P. Senet, C. Van Alsenoy, Int. J. Quantum Chem. 101 (2005) 535. M.J. Frisch et al., Gaussian, Inc., Wallingford, CT, 2004, GAUSSIAN 03. R.A. Teichman III, E.R. Nixon, J. Mol. Spectrosc. 54 (1975) 78. Y. Noda, K. Masumoto, S. Ohba, Y. Saito, K. Toriumi, Y. Iwata, I. Shibuya, Acta Crystallogr. Sect. C Cryst. Struct. Commun. 43 (1987) 1443. B.L. Wang, S. Nagase, J.J. Zhao, G.H. Wang, J. Phys. Chem. C 111 (2007) 4956. H.-S. Wu, F.-Q. Zhang, X.-H. Xu, C.-J. Zhang, H.J. Jiao, J. Phys. Chem. A 107 (2003) 204. A. Costales, M.A. Blanco, E. Francisco, A.M. Pendás, R. Pandey, J. Phys. Chem. B 110 (2006) 4092. J.J. Zhao, B.L. Wang, X.L. Zhou, X.S. Chen, W. Lu, Chem. Phys. Lett. 422 (2006) 170. J.M. Matxain, J.M. Mercero, J.E. Fowler, J.M. Ugalde, J. Am. Chem. Soc. 125 (2003) 9494. F.B. van Duijneveldt, J.G.C.M. Van Duijneveldt-van de Rijdt, J.H. van Lenthe, Chem. Rev. 94 (1994) 1873. W.D. Knight, K. Clemenger, W.A. de Heer, W.A. Saunders, M.Y. Chou, M.L. Cohen, Phys. Rev. Lett. 52 (1984) 2141. P. Karamanis, D. Bégué, C. Pouchan, J. Chem. Phys. 127 (2007) 094706. P. Karamanis, G. Maroulis, C. Pouchan, J. Chem. Phys. 124 (2006) 071101. J.L. Brédas, C. Adant, P. Tackx, A. Persoons, B.M. Pierce, Chem. Rev. 94 (1994) 243. B. Champagne, et al. , J. Chem. Phys. 109 (1998) 10489. B. Champagne, et al. , J. Phys. Chem. A 104 (2000) 4755.