JChemPhys 123 244303 cu doped ge cluster

A computational investigation of copper-doped germanium and germanium clusters by the density-functional theory Jin Wang and Ju-Guang Han Citation: J Chem Phys 123, 244303 (2005); doi: 10.1063/1.2148949 View online: http://dx.doi.org/10.1063/1.2148949 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v123/i24 Published by the American Institute of Physics Additional information on J Chem Phys Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors Downloaded 10 Jun 2013 to 134.58.49.37 This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions THE JOURNAL OF CHEMICAL PHYSICS 123, 244303 ͑2005͒ A computational investigation of copper-doped germanium and germanium clusters by the density-functional theory Jin Wanga͒,b͒ Department of Chemistry, University of Guelph, Guelph N1G 2W1, Ontario, Canada Ju-Guang Hanb͒,c͒ Department of Chemistry, Jackson State University, Jackson, Mississippi 39217 ͑Received October 2005; accepted 10 November 2005; published online 23 December 2005͒ The geometries, stabilities, and electronic properties of Gen and CuGen ͑n = – 13͒ clusters have been systematically investigated by using density-functional approach According to optimized CuGen geometries, growth patterns of Cu-capped Gen or Cu-substituted Gen+1 clusters for the smallor middle-sized CuGen clusters as well as growth patterns of Cu-concaved Gen or Ge-capped CuGen−1 clusters for the large-sized CuGen clusters are apparently dominant The average atomic binding energies and fragmentation energies are calculated and discussed; particularly, the relative stabilities of CuGe10 and Ge10 are the strongest among all different sized CuGen and Gen clusters, respectively These findings are in good agreement with the available experimental results on − and Ge10 clusters Consequently, unlike some transition metal ͑TM͒Si12, the hexagonal CoGe10 prism CuGe12 is only low-lying structure; however, the basketlike structure is located as the lowest-energy structure Different from some TM-doped silicon clusters, charge always transfers from copper to germanium atoms in all different sized clusters Furthermore, the calculated highest occupied molecular orbital and lowest unoccupied molecular orbital ͑HOMO-LUMO͒ gaps are obviously decreased when Cu is doped into the Gen clusters, together with the decrease of HOMO-LUMO gaps, as the size of clusters increases Additionally, the contribution of the doped Cu atom to bond properties and polarizabilities of the Gen clusters is also discussed © 2005 American Institute of Physics ͓DOI: 10.1063/1.2148949͔ I INTRODUCTION In recent years, clusters of group-14 elements have attracted attention because they are important for fine processing of semiconductors and synthesizing novel materials Previous investigations indicate that the pure silicon is unfavorable for forming large-sized clusters and bulk solids; however, the encapsulation of transition metal in the largesized silicon cluster contributes to enhancing stability of pure silicon clusters and simultaneously exhibiting many magic behaviors, e.g., sized selectivities, different frontier orbital properties, and polarizabilities.1–15 Compared with silicon and transition-metal-͑TM͒ doped silicon clusters, present investigations on the germanium clusters mainly focus on different sized pure germanium clusters or halogen-doped germanium clusters.16–22 For different sized Gen clusters, many properties such as geometries, binding energies, ionization potentials, electron affinities, dipole polarizabilities, etc., are calculated by using different theoretical methods On the other hand, atomization enthalpies and enthalpies of formation for the different sized germanium clusters are measured through Knudson effusion mass spectrometry.23 The eleca͒ Electronic mail: jwang02@uoguelph.ca On leave from National Synchrotron Radiation Laboratory, University of Science and Technology of China, Hefei 230026, People’s Republic of China c͒ Author to whom correspondence should be addressed FAX: 86-5515141078, Electronic mail: jghan@ustc.edu.cn b͒ 0021-9606/2005/123͑24͒/244303/12/$22.50 tronic binding energies of the Gen ͑n = – 32͒ clusters and halogen-doped Gen ͑n = – 20͒ clusters had been measured by aid of photoelectron spectroscopy experiment.24 In addition, the photoionization thresholds of the Gen ͑n = – 57͒ and Snn ͑n = – 41͒ clusters were examined by laser photoionization coupled with reflectron time-of-flight mass spectrometry;25 their experimental results indicated that there was a major maximum of ionization potential ͑IP͒ for the Gen clusters at n = 10 and a rapid decrease of IP at 15ഛ n Ͻ 26 Although the Gen clusters were studied in detail through the theoretical and experimental methods, a few theoretical investigations on the transition-metal-doped germanium clusters are reported Ab initio pseudopotential planewave method was employed to investigate the encapsulated caged TMGen ͑n = 14– 16͒ clusters.26 Their theoretical results revealed that growth behavior of metal-encapsulated germanium clusters was different from that of metal-encapsulated silicon clusters and had large highest occupied molecular orbital-lowest unoccupied molecular orbital ͑HOMOLUMO͒ gaps, etc Additionally, they also adopted identical calculation method to study structures of ZnGe12 and CdSn12,27 and pointed out that these clusters had perfect icosahedral symmetry and large HOMO-LUMO gap of about eV So far, there is no theoretical investigation on the Cudoped germanium clusters reported In this work, a detailed investigation on the structures, stabilities, and bonding properties of the Cu-doped germanium clusters are calculated at 123, 244303-1 © 2005 American Institute of Physics Downloaded 10 Jun 2013 to 134.58.49.37 This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions 244303-2 J Wang and J.-G Han UB3LYP/LanL2DZ level In order to examine the effect of the doped Cu atom to the germanium clusters, geometry optimizations of the pure germanium clusters are also calculated by using identical methods and basis sets II COMPUTATIONAL DETAILS The geometry optimizations of the Gen and CuGen ͑n = – 13͒ clusters with spin configurations considered are performed by using density-functional theory ͑DFT͒ with the unrestricted B3LYP exchange-correlation potential28,29 and an effective core potential LanL2DZ basis sets The standard LanL2DZ basis sets,30 which provide an effective way to reduce difficulties in calculations of two-electron integrals caused by heavy transition-metal atom, are employed In previous paper, the LanL2DZ basis sets of effective core potential ͑ECP͒ theory are proven to be reliable for the geometries, stabilities, and electronic properties of the TM-Sin and Gen systems.31–35,21 In order to test the reliability of our calculation, the Cu2 dimer is calculated The theoretical results indicate that the singlet Cu2 dimer is the most stable state, and that the Cu–Cu bond length ͑2.259 Å͒ and vibrational frequency ͑256 cm−1͒ obtained by using LanL2DZ basis sets are in good agreement with the experimental values of 2.22 Å and 266 cm−1, respectively.36,37 All theoretical calculations are performed with 38 GAUSSIAN-03 program package In this paper, equilibrium geometries of the Gen clusters are optimized first On the basis of the optimized Gen geometries, different evolution patterns for determining the different sized CuGen isomers, i.e., Cu-capped, Cu-substituted, and Cu-concaved patterns as well as Ge-capped pattern, are taken into accounts In order to test the basis sets, the stabilities of CuGen clusters are calculated and discussed first The basis sets labeled GEN, which are the combinations of LanL2DZ and 6-311+ G͑2d͒ basis sets, are employed for the Cu and Ge atoms, respectively The calculated results by using the GEN basis sets indicate that the total energy of every isomer decreased; however, the energetic ordering of the competitive isomers for a definite-sized CuGen clusters is essentially unchanged as compared to the results calculated by using the LanL2DZ basis sets Consequently, the LanL2DZ basis sets are reliable and accurate enough to be applied to describe the properties of the CuGen and Gen clusters in this paper Furthermore, all the calculated results in this paper are calculated at the UB3LYP/LanL2DZ level On the basis of the calculated structures with different spin states considered, the optimized results show that the most stable structures of the Cu-doped germanium clusters are spin doublet states in all isomers However, in the case of the pure germanium clusters, the most stable state of Ge2 dimer is triplet spin state and the most stable state of other different sized germanium clusters corresponds to singlet spin state in all isomers III RESULTS AND DISCUSSIONS A Growth behavior of different sized germanium clusters Different spin states of the Ge2 dimer are considered, and the optimized results indicate that the triplet spin state is J Chem Phys 123, 244303 ͑2005͒ lower in total energy than the singlet spin state by 0.643 eV Therefore, the 3⌺−g state is found to be ground state for the Ge2 dimer This result is in agreement with previous calculation by using different density-functional methods.18,39 The open triangular structure, the closed triangular structure, and the linear structure of the Ge3 isomer are considered; the most stable geometry corresponds to an open triangular structure with spin singlet state In addition, the closed triangular structure with spin triplet state can be also optimized to be a local minimum; however, its total energy is higher by 0.259 eV than that of the open triangular structure with singlet spin state According to previous calculation on the Ge3, the electronic state of 1A1 is assigned as the lowest-energy state which is in agreement with our calculated results.18,39 As shown in Fig 1, two stable Ge4 structures, i.e., a planar rhombus 4a and a pyramid 4b, can be generated by capping Ge atom on the surface site of the stable Ge3 frame and optimized to be a local minimum Furthermore, the planar rhombus 4a structure is obviously more stable than the pyramid 4b structure because the total energy of the former is distinctly lower than that of the latter by 2.122 eV Therefore, it is confirmed that the lowest-energy Ge4 cluster is a planar rhombus structure with electronic state of 1A1 which is similar to the lowest-energy Si4 isomer and the previous calculations.18,39 Two stable 5a and 5b Ge5 structures can be described as the distorted trigonal bipyramid and nonplanar pentagon, which are generated by capping one Ge on the Ge4 4a As seen from the total energy, the distorted trigonal bipyramid 5a isomer is more stable than the nonplanar pentagon 5b isomer As seen from our optimized geometries, the lowestenergy Ge5 isomer is a distorted trigonal bipyramid D3h structure having 1A1Ј character which supports the previous calculated results by using different density-functional methods.39 Additionally, the previous ab initio pseudopotential configuration-interaction ͑CI͒ calculation suggests that the triplet spin state of the standard trigonal bipyramid with D3h symmetry is a ground state.40 However, our calculations not support this earlier calculation because this structure is optimized to be an unstable structure with two imaginary frequencies In analogy to the Ge5 isomer, the out-of-plane edge-capped patterns can also produce the stable Ge6 structures: one is a bicapped rhombic pyramid 6a and the other is a boatlike 6b structure A stable Ge6 6a isomer is described as one Ge atom being out-of-plane edge-capped between two Ge atoms of bent rhombus As seen from Table I, the total energy of the bicapped rhombic pyramid structure is lower than that of the boatlike structure by 0.783 eV, indicating that the stability of the bicapped rhombic pyramid is stronger as compared to that of the boatlike structure Furthermore, the electronic state of the lowest-energy Ge6 6a cluster is shown to be 1A in our calculation, which is slightly different from the previous report, 1A1, of the identical structure with C2v structure;18,39 however, its total energy of the bicapped rhombic pyramid C2v isomer ͑−22.625 642 hartree͒ is almost the same as that of the identical structure with C1 symmetry ͑−22.625 643 hartree͒ in this work According to vibrational frequency analysis, three Ge7 structures are proven to be the stable structures One is a Downloaded 10 Jun 2013 to 134.58.49.37 This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions 244303-3 Copper-doped germanium clusters J Chem Phys 123, 244303 ͑2005͒ FIG All the equilibrium Gen ͑n = – 13͒ clusters The stars show the lowest-energy Gen ͑n = – 13͒ structures pentagonal bipyramid Ge7 7a cluster, which can be produced by two Ge atoms being symmetrically face capped on the top of the bent rhombus of the Ge5 5a cluster The other belongs to a typical multirhombus 7b structure, which is composed of the planar or bent rhombus A stable 7c structure is generated with one Ge atom being face capped on the top of the boatlike Ge6 6b cluster As illustrated from Table I, the pentagonal bipyramid, which is the lowest-energy geometry as compared to the multirhombus structure, is still a dominant structure Therefore, our calculated results are in good agreement with the previous full-potential linear muffin-tin orbital molecular-dynamics ͑FP-LMTOMD͒ method calculations.41 Furthermore, one pentagonal bipyramid D5h isomer with electronic state of 1A1Ј is the lowest-energy structure Similar to the Ge7 cluster, the pentagonal bipyramid Si7 cluster was also proven to be ground state in the previous calculation.42 As far as the Ge8 cluster is concerned, one structure, which is obtained from the pentagonal bipyramid Ge7 7a, is proven to be stable structure Another face-capped pentagonal pyramid or multirhombus ͑1Ge–2Ge–3Ge–4Ge rhombus and 5Ge–6Ge–7Ge–8Ge rhombus͒ 8b cluster is verified to be an equilibrium structure; however, its stability is weaker than the 8a due to its total energy that is slightly higher than that of the 8a isomer Previous FP-LMTO calculation pointed out that the identical 8b cluster with Cs symmetry was the most stable structure.41 However, our calculation proves that the identical structure is not the most stable structure because its total energy is slightly higher than that of the 8a cluster by 0.09 eV On the basis of the Ge7 7c isomer, one stable 8c structure is formed with respect to one Ge atom being the face capped on the bottom of the Ge7 7c isomer As illustrated in Table I, the total energy of the face-capped hexagonal bipyramid is higher than those of other isomers Hence, the lowest-energy Ge8 8a isomer can be described as having A character When the size of Gen clusters is up to 9, four kinds of structures can be verified to be the minima The lowestenergy 9a and low-lying 9c and 9d isomers are described as one Ge being face capped on the top, bottom, and side of the multirhombus Ge8 isomer, respectively A stable 9b structure is generated by capping two Ge atoms on the surface sites of the Ge7 7a isomer As compared with the low-lying Ge9 9cstructure, it should be mentioned that the analogous structure of the Si9 isomer is the most stable structure.42 In this work, four kinds of Ge10 clusters are found as minima The lowest-energy 10a structure with electronic Downloaded 10 Jun 2013 to 134.58.49.37 This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions 244303-4 J Chem Phys 123, 244303 ͑2005͒ J Wang and J.-G Han TABLE I Geometries, and total energies of CuGen and Gen ͑n = – 13͒ clusters Sym means point-group symmetry State denotes electronic state; RCu–Ge and RGe–Ge represent the shortest Cu–Ge and Ge–Ge bond lengths, respectively ET denotes the total energies of different CuGen and Gen isomers ⌬E means relative energies of every isomer compared to that of the lowest-energy isomer for identical-sized cluster RGe–Ge ͑Å͒ ET ͑hartree͒ 2.524 2.488 −203.665 109 ⌬E ͑eV͒ 2.566 2.429 2.604 2.458 −211.228 159 −211.227 946 C1͑aЈ͒ C2v͑bЈ͒ Cs͑cЈ͒ A A2 AЈ 2.617 2.423 2.497 2.58 2.528 2.523 −215.023 147 −215.007 973 −215.031 680 Cs͑aЈ͒ C1͑bЈ͒ C1͑cЈ͒ C1͑dЈ͒ C1͑eЈ͒ AЉ A A A A 2.489 2.578 2.531 2.593 2.495 2.682 2.579 2.687 2.719 2.506 −218.813 250 −218.812 255 −218.796 507 −218.806 640 −218.795 554 A AЉ A A 2.555 2.576 2.509 2.497 2.549 2.578 2.574 2.537 −222.578 322 −222.588 611 −222.595 596 −222.595 796 0.475 0.195 0.005 Ge7 0.115 0.3 0.351 0.147 Ge8 CuGe8 CuGe9 CuGe10 CuGe11 CuGe12 CuGe13 0.006 0.232 0.645 0.027 0.456 0.179 0.482 2.549 2.415 2.424 2.414 2.469 2.577 2.699 2.541 2.567 2.642 −226.371 077 −226.364 274 −226.362 396 −226.369 904 −226.375 306 C1͑aЈ͒ C1͑bЈ͒ C1͑cЈ͒ A A A 2.404 2.504 2.547 2.647 2.627 2.575 −230.177 727 −230.160 815 −230.142 760 0.46 0.951 D4d͑aЈ͒ C1͑bЈ͒ C1͑cЈ͒ C1͑dЈ͒ A1 A A A 2.551 2.626 2.499 2.474 2.786 2.59 2.587 2.581 −234.001 926 −233.977 704 −233.922 324 −233.949 378 0.659 2.166 1.43 C1͑aЈ͒ C1͑bЈ͒ C2v͑cЈ͒ C1͑dЈ͒ A A A2 A 2.543 2.554 2.468 2.508 2.619 2.621 2.627 2.552 −237.765 713 −237.764 634 −237.762 001 −237.746 072 0.029 0.101 0.534 C1͑aЈ͒ C1͑bЈ͒ Cs͑cЈ͒ C1͑dЈ͒ C1͑eЈ͒ C1͑fЈ͒ A A AЉ A A A 2.643 2.58 2.53 2.55 2.494 2.682 2.682 2.554 2.634 2.556 2.513 2.538 −241.536 991 −241.531 088 −241.534 156 −241.543 187 −241.519 539 −241.539 065 0.169 0.329 0.246 C1͑aЈ͒ C1͑bЈ͒ C1͑cЈ͒ C1͑dЈ͒ 2.512 2.6 2.692 2.705 2.499 2.5 2.557 2.519 −245.282 702 −245.320 642 −245.313 211 −245.310 804 1.032 2 A A A A state of 1A1 can be depicted as tetracapped trigonal prism structure Another stable-caged 10b isomer can be described as irregular pentagonal prism structure Different from the two equilibrium structures above, the novel basketlike 10c Ge5 Ge8 A A AЉ AЈ A AЈ1 A 2.587 2.45 −18.838 419 −18.826 697 0.319 C1͑a͒ C1͑b͒ A A 2.601 2.514 −22.625 643 −22.596 841 0.784 D5h͑a͒ C2v͑b͒ Cs͑c͒ A1Ј A1 A1 2.789 2.519 2.507 −26.419 340 −26.386 127 −26.409 663 0.904 0.263 1 A AЈ A1 2.533 2.572 2.576 −30.177 205 −30.174 003 −30.170 533 0.087 0.182 0.288 C1͑aЈ͒ C1͑bЈ͒ Cs͑cЈ͒ Cs͑dЈ͒ C1͑eЈ͒ 2.122 A A −15.057 266 −14.979 266 D3h͑a͒ C1͑b͒ C1͑aЈ͒ C1͑bЈ͒ C1͑aЈ͒ Cs͑bЈ͒ C1͑cЈ͒ C1͑dЈ͒ 2.563 2.843 −207.436 551 −207.443 949 −207.433 372 CuGe7 A1 A C2v͑a͒ C1͑b͒ 2.434 2.457 2.577 −11.250 610 Ge4 2.456 2.421 2.604 CuGe6 2.378 C 2v B2 A1 A A1 Ge3 C2v͑aЈ͒ C2v͑bЈ͒ C1͑CЈ͒ CuGe5 −7.460 758 CuGe3 2.548 0.201 ⌺−g Dϱh 2 ET ͑hartree͒ Ge2 C 2v B1 RGe–Ge ͑Å͒ Sym CuGe2 ⌬E ͑eV͒ State Cluster Sym CuGe4 State RCu–Ge ͑Å͒ Cluster Ge9 Ge10 C1͑a͒ Cs͑b͒ C2v͑c͒ 1 1 Cs͑a͒ Cs͑b͒ C1͑c͒ Cs͑d͒ AЈ AЈ A AЈ 2.6 2.518 2.57 2.489 −33.976 275 −33.940 289 −33.975 948 −33.950 273 0.979 0.009 0.708 C3v͑a͒ C1͑b͒ Cs͑c͒ C1͑d͒ A1 A AЈ A 2.626 2.553 2.581 2.561 −37.777 426 −37.758 821 −37.715 344 −37.732 405 0.506 1.689 1.225 A 2.598 −41.536 602 1 Ge11 C1 Ge12 C1͑a͒ C1͑b͒ C1͑c͒ A A A 2.562 2.58 2.489 −45.307 604 −45.299 904 −45.298 562 C1 2.503 −49.062 096 0.21 0.246 0.643 0.112 Ge13 A 0.202 0.268 and polyhedral 10d structures are born However, the total energy of the 10c isomer is obviously higher than those of the 10a and 10b isomers by 1.688 and 1.183 eV, respectively Downloaded 10 Jun 2013 to 134.58.49.37 This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions 244303-5 Copper-doped germanium clusters Compared to other sized Gen clusters, one Ge11 structure can be found as minimum and this structure can be regarded as one Ge atom being face capped the top of the tetracapped trigonal prism Ge10 cluster When the size of Gen cluster keeps increasing to n = 12, three different geometries can be found as minima in present calculation The basketlike structure being optimized to be an equilibrium structure is more stable than another cagelike stable structure which is composed of the double hexagonal Ge6 ring As far as the Ge13 cluster is concerned, one basketlike 13a structure is found as minimum and this geometry appears floppy In conclusion, for the growth pattern of the Gen clusters, the in-plane-capped Ge atom only appears in the very smallsized clusters and the out-of-plane edge-capped or facecapped Ge atom is dominant in the different sized Gen clusters B Growth behavior of different sized copper-doped germanium clusters The possible CuGe2 geometries such as two linear isomers and a triangular structure are considered; only the C2v CuGe2 structure with doublet spin configuration involved in the Cu atom directly being capped on the Ge2 or substitution of the central Ge in the Ge3 cluster is optimized to be the most stable structure The identical structure with spin quartet state is also optimized to be the stable structure; however, its total energy is obviously higher than that of spin doublet state by 1.089 eV Therefore, the electronic state of the lowest-energy CuGe2 isomer can be described as 2B1 Three kinds of CuGe3 clusters can be optimized to be the minima at the UB3LYP/Lan2DZ level Interestingly, two rhombic structures, which have identical C2v symmetry, can be found as the different isomers with different Ge–Cu–Ge angles, indicating that the two rhombic structures are generated from different precursory molecules The Ge–Ge–Ge bond angle ͑130.7°͒ of the rhombus 3aЈ isomer, generated from substitution of Ge4 4a rhombus by Cu, is much larger than that of the Ge3 cluster However, the Ge–Ge–Ge bond angle of another rhombus 3bЈ isomer, described as Cu being capped on the Ge3 isomer, is very close to that of the Ge3 structure When one Ge atom in the Ge4 4b is substituted by Cu, a new pyramid 3cЈ structure is formed Furthermore, the rhombus 3bЈ CuGe3 structure is more stable than the pyramid 3cЈ structure in that the total energy of the former is lower than that of the latter On the basis of the most stable CuGe3 3bЈ structure, the spin quartet state is considered and optimized to be a transition state with one imaginary frequency Therefore, the 3bЈ isomer is the lowest-energy isomer which has electronic state of 2A1 According to the discussion above, it is concluded that the dominant growth pattern for the small CuGen isomers is still described as Cu being capped on the Gen clusters Two stable CuGe4 structures are considered One is obtained from the Cu substitution of the apical Ge atom in the Ge5 5a isomer The new stable CuGe4 4bЈ isomer is yielded with the top Ge atom in the Ge5 5b cluster being substituted by Cu The former is more stable than the latter; therefore, the 4aЈ is the lowest-energy isomer J Chem Phys 123, 244303 ͑2005͒ Three kinds of CuGe5 isomers can be generated as the minima Similarly, two stable CuGe5 structures can be viewed as being generated from the Ge6 6a cluster An analogous bicapped rhombic pyramid 5aЈ structure is formed when the bottom Ge atom in the Ge6 6a isomer is substituted by copper In addition, the boatlike Ge6 structure is distorted into a planar CuGe5 5bЈ isomer when one Ge atom in the Ge6 6b isomer is substituted by Cu However, this stable structure is much higher in total energy than the 5aЈ isomer One notes that when the top Ge atom in the Ge6 6a isomer is substituted by a copper another different 5cЈ isomer is obtained as compared to the 5aЈ isomer However, the total energy of the 5cЈ cluster is obviously lower than that of the 5aЈ cluster by 0.232 eV and it corresponds to the lowestenergy cluster with electronic state of 2AЈ Therefore, the different substituted positions of the Cu atom in the Gen clusters lead to the different energetic CuGen−1 isomers As mentioned above, when copper replaces different Ge atom in the multirhombus Ge7 7b, different energetic CuGe6 ͑6aЈ, 6bЈ, and 6cЈ͒ isomers can be produced Moreover, the substitution of the surface Ge atom by Cu in the Gen clusters causes the distortion of geometry In addition, it should be pointed out that the most stable copper-doped Gen clusters may not be generated directly from the most stable Gen clusters and it depends on the substituted pattern or position For example, on the basis of the lowest-energy 7a structure, only the substitution of the bottom Ge atom in the Ge7 7a cluster can result in the stable 6dЈ CuGe6 structure; however, this isomer is not the lowest-energy CuGe6 isomer Besides the mentioned equilibrium CuGe6 structures, a new stable 6eЈ structure can be born and is described as Cu directly being capped on the boatlike Ge6 6b geometry For the lowestenergy CuGe6 6aЈ structure, its electronic state can be assigned as 2AЉ On the basis of the Ge8 8a cluster, when one Ge atom in bent rhombus is substituted by Cu, one stable 7aЈ structure is yielded Furthermore, when different Ge atom in the facecapped pentagonal pyramid or multirhombus Ge8 8b cluster is substituted by Cu, different energetic 7bЈ and 7cЈ isomers can be yielded The calculated total energy reveals that the replacement of the bottom Ge atom is superior to the top Ge atom in that the stability of the 7cЈ isomer is stronger than that of the 7bЈ isomer Compared with other CuGe7 isomers, the most stable 7dЈ structure is formed when the top Ge atom in the face-capped hexagonal bipyramid Ge8 8c isomer is substituted by Cu In addition, the spin quartet state of the CuGe7 7dЈ structure is also considered and optimized to be a stable structure; furthermore, its total energy of spin quartet state is higher than that of spin doublet state by 0.977 eV It should be mentioned that the electronic state of the lowestenergy doublet CuGe7 cluster is 2A Beginning from the CuGe8 cluster, a copper-concaved structure appears; however, Cu is not totally encapsulated in the Ge8 cage It can be seen that the concaved structure can keep the analogous framework as the original clusters; however, the surface-inserted or extracapped Ge atom on the small CuGen−1 clusters causes obvious deformation of the framework of original clusters Although the concaved CuGe8 8bЈ structure beings to be formed, it does not belong Downloaded 10 Jun 2013 to 134.58.49.37 This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions 244303-6 J Wang and J.-G Han to the lowest-lying isomer which is reflected from its total energy As observed from the optimized structures, the stable CuGe8 8cЈ, CuGe8 8aЈ, and CuGe8 8dЈ isomers are formed when the different Ge atoms in Ge9 9a, Ge9 9b, and Ge9 9d are substituted by Cu, respectively In addition, when one Ge atom in the Ge9 9c isomer is replaced by Cu, the most stable CuGe8 8eЈ isomer can be obtained and optimized to be the lowest-energy isomer Three different stable CuGe9 isomers are located at UB3LYP/Lan2LDZ level On the basis of the Ge9 9c cluster, the different Cu added patterns, i.e., the capped and concaved patterns, can form different energetic CuGe9 9aЈ and 9bЈ isomers, respectively However, total energy of the 9bЈ cluster is obviously higher than that of the 9aЈ cluster, indicating that Cu is easily concaved into the Gen clusters at the beginning of the CuGe9 cluster Different from the CuGe8 cluster, the dominant CuGe9 structure varies from the surface-inserted Cu atom to the Cu-concaved structure As illustrated in Table I, the total energy of the concaved 9aЈ structure is lower by 0.95 eV than that of the surface-inserted 9cЈ structure Therefore, the 9aЈ isomer is selected as the lowest-energy isomer In analogy to the CuGe9, although the surface-inserted CuGe10 structure is still the stable structure; however, it is no longer candidate for the lowest-energy state because of its higher total energy as compared to other isomers In this sized clusters, two different encapsulated structures are considered One is a typical multirhombus-caged D4d 10aЈ structure, and the other 10bЈ cluster being described as the Cuencapsulated face-to-face pentagonal Ge10 cage is generated from the Cu concaved in the Ge10 10b cluster The surfacecapped CuGe10 10cЈ cluster is formed with Cu being absorbed on the surface of the basketlike Ge10 10c cluster Obviously, stability of the 10cЈ isomer is weaker than those of the other two encapsulated structures due to its relatively higher total energy on the potential-energy surface Additionally, when one Ge atom is substituted by Cu in the Ge11 isomer or Cu is inserted in the Ge10 10a isomer, a stable CuGe10 10dЈ structure is found and its total energy is obviously higher than that of the encapsulated structure Therefore, the encapsulated 10aЈ isomer is the lowest-energy isomer with electronic state of 2A1 Although only one stable Ge11 cluster is yielded, the copper-doped Ge11 cluster can produce more stable isomers or all the isomers are generated from the small-sized CuGen clusters by addition of an extra Ge atom For example, on the basis of the face-to-face pentagonal CuGe10 10bЈ cluster, the lowest-energy CuGe11 11aЈ structure can be formed with one Ge atom being face capped the top of the pentagon Ge5 clusters In addition, on the basis of the lowest-energetic multirhombus 10aЈ structure, one stable CuGe11 11bЈ isomer which has higher energy than 11aЈ is produced when the 11th Ge atom is edge capped on the rhombus Ge4 cluster and the upper bent rhombus in the CuGe10 10aЈ cluster is destroyed Analogous to the 11aЈ structure, a face-to-face pentagon 11cЈ isomer is optimized to be stable structure; however, the face-to-face pentagon 11cЈ isomer is staggered as compared to the parallel face-to-face pentagon 11aЈ cluster According to the calculated total energy, the stability of the J Chem Phys 123, 244303 ͑2005͒ 11cЈ isomer is weaker than that of the 11aЈ isomer; a new stable 11dЈ isomer, which is higher in total energy than the 11aЈ cluster, is yielded when the 11th Ge atom is capped the surface site of the CuGe10 10bЈ Hence, it is exhibited that the unfavorable additional pattern on the most stable smallsized CuGen structures can still yield the relatively highenergetic large-sized isomers Guided by the growth pattern of the CuGe11 isomer, six different CuGe12 isomers are obtained One equilibrium 12aЈ structure is formed with one Ge atom being edge capped the top of the upper hexagon 11aЈ cluster Additionally, when the Ge atom is edge capped on the low pentagon in the 11dЈ cluster, a new equilibrium 12bЈ structure is yielded The lowlying 12bЈ structure is also thought as Cu being encapsulated into the hexagonal-prism-caged Ge12 12b structure Surprisingly, the total energy of the 12bЈ cluster is higher than that of the 12aЈ and the stability of the 12bЈ isomer is weaker than that of the 12aЈ cluster However, for the TM-doped Si12 cluster, the analogous concaved hexagonal prism structure is usually the most stable structure.4,31,43–45 This finding indicates that the TM-doped Gen clusters have different growth behavior as compared to the TM-doped Sin clusters On the basis of the basketlike Ge12 12a structure, one stable 12cЈ structure can be formed with Cu being directly encapsulated in the center site of the Ge12 cage As shown in Fig 2, the Cu-doped Ge12 cage does not distort the Ge12 frame and it changes the stability of the Ge12 structure On the basis of the Ge12 12c structures, the high-energetic Cu-doped CuGe12 12eЈ isomer is formed when Cu is directly encapsulated in the Ge12 cage Except for the encapsulated hexagonal prism 12bЈ isomer, another low-lying encapsulated bicapped pentagonal prism 12fЈ cluster is also obtained Surprisingly, the most stable 12dЈ CuGe12 structure is not an encapsulated structure, but is a concaved basketlike structure Similar to our results, Kawamura et al pointed out that the basketlike TiSi12 isomer has lower energy than that of the cage isomer.46 On the basis of the pure Ge13 cluster, one equilibrium CuGe13 13aЈ structure is generated when Cu is directly encapsulated in the Ge13 cage As observed in Fig 2, Cu doped in the center site of the Ge13 cage keeps the analogous framework as the Ge13 cage and does not obviously distort the geometry of the Ge13 cage Analogous to CuGe12, the lowestenergy CuGe13 13bЈ isomer is optimized to be an irregular basketlike structure and it is more stable than the 13aЈ isomer In addition, two different CuGe13 13cЈ and 13dЈ isomers are generated when one Ge atom is edge capped on the CuGe12 12fЈ and CuGe12 12bЈ isomers Furthermore, the CuGe13 13cЈ and CuGe13 13dЈ clusters are higher in the total energies than the 13bЈ cluster In conclusion, three kinds of growth patterns, i.e., Cucapped or concaved Gen clusters, Cu-substituted Gen+1 clusters, as well as Ge-capped CuGen−1 clusters, are considered for the different sized CuGen clusters For the CuGe2 and CuGe3 clusters, Cu directly capping on the identical-sized Gen clusters are dominant growth pattern However, for other small-sized CuGen ͑n = – 8͒ clusters, Cu directly substituted Ge atom of the Gen+1 clusters to form the CuGen clusters is Downloaded 10 Jun 2013 to 134.58.49.37 This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions 244303-7 Copper-doped germanium clusters J Chem Phys 123, 244303 ͑2005͒ FIG All the equilibrium CuGen ͑n = – 13͒ clusters The starts show the lowest-energy CuGen ͑n = – 13͒ structures dominant growth pattern When the size of CuGen exceeds 8, two dominant different growth patterns for the middle-sized or large-sized clusters are considered One is Cu directly being inserted into the Gen framework and the other is the Ge edge capped on the CuGen−1 framework It can be ex- pected that Ge face-capped pattern possibly becomes the dominant pattern with the growth of the CuGen clusters when the size of the CuGen clusters is larger than 13 Additionally, it should be mentioned that Cu-doped Gen structures contribute to stabilizing Gen clusters Downloaded 10 Jun 2013 to 134.58.49.37 This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions 244303-8 J Wang and J.-G Han J Chem Phys 123, 244303 ͑2005͒ changed As observed from HOMO of the CuGe6 6eЈ cluster, the bonding of 1Ge–6Ge and 3Ge–4Ge appears ␲-type bond In addition, the delocalized d-p ␲-type bond is formed between Cu and 1Ge–3Ge–4Ge–6Ge, and it indicates that the capped Cu atom in the Gen clusters strengthens ␲-type bond delocalization As discussed before, when the seventh Ge atom in the Ge7 ͑7a͒ is substituted by Cu, the stable CuGe6͑dЈ͒ isomer is formed Although the geometry is not distinctly distorted, the bonding property is obviously changed As seen from HOMO of the Ge7 ͑7a͒, except for the second Ge and fourth Ge atom, all the other Ge atoms participate in formation of the ␴-type bond However, for the substituted CuGe6͑dЈ͒, the second and fourth Ge atoms begin to form the delocalized ␲-type bond with other Ge atoms As far as the large-sized Cu-concaved Gen clusters are concerned, the ␴-type bond types among Ge atoms, which are distributed beside Cu, usually change to be the delocalized ␲-type bonds For example, as seen from HOMO of the basketlike CuGe12 12cЈ, the obvious delocalized ␲-type bond is formed between Cu and adjacent Ge atoms as compared to the identical Ge12 12a isomer FIG Contour maps of the HOMOs of the selected CuGen and Gen clusters C Effects of the doped Cu to bonding properties of Gen clusters As seen from the properties of the HOMO of the Ge2 dimer ͑Fig 3͒, the ␲-type bond is formed between the Ge and Ge atoms However, when Cu is capped on the Ge2 molecule, the ␲-type bond is changed to be the ␴-type bond As observed from the HOMO of the Ge3 isomer, the delocalized ␲-type orbitals extend over all the three Ge atoms As mentioned above, two kinds of the CuGe3 geometries, which have identical C2v symmetry, can be generated However, their shapes of the HOMO and bonding properties are obviously different For the C2v͑aЈ͒ CuGe3, its HOMO obviously corresponds to a ␴-type bond with mixed Cu d characters and the density around the second Ge atom is very low However, for the C2v͑bЈ͒ CuGe3, its HOMO corresponds to the delocalized ␲-type bond on the Ge3 unit with small admixture of the Cu d characters According to the bonding properties of the two CuGe3 isomers, it indicates that the different evolution modes exist As far as the C2v͑aЈ͒ CuGe3 isomer is concerned, it is generated from the substitution of Cu on the Ge4 isomer But, the C2v͑bЈ͒ CuGe3 isomer is formed by capping Cu on the Ge3 C2v isomer As observed from HOMO of the Ge4 ͑4a͒, the 2Ge–4Ge bond appears the ␲-type bond and 1Ge–3Ge bond appears the ␴-type bond When the fourth Ge atom is substituted by Cu, the ␲-type bond of 2Ge–4Ge disappears and the d-p ␴-type bond of 1Ge–Cu–3Ge is formed For the boatlike Ge6 6b isomer, the bonds of 1Ge–6Ge and 3Ge–4Ge are obvious ␴-type bond with p subshell in character; however, the density populated on the second Ge and fifth Ge atoms is very low When Cu is capped on the top of the Ge6 cluster, the bonding properties are distinctly D Averaged binding energy, fragmentation energy, and embedding energy of the Gen and CuGen clusters It is known that the relative stability of the different sized clusters can be predicted by calculating the averaged binding energy and fragmentation energy Moreover, because the doped Cu on Gen clusters can also influence the relative stabilities of certain sized Gen clusters, the calculated results on the averaged binding energy and fragmentation energy of the Gen and CuGen clusters provide an interpretation of the above influence The averaged binding energies and fragmentation energies for the Gen clusters can be defined as the following formula: Eb͑n͒ = ͓nET͑Ge͒ − ET͑Gen͔͒/n, D͑n,n − 1͒ = ET͑Gen−1͒ + ET͑Ge͒ − ET͑Gen͒, where ET͑Gen−1͒, ET͑Ge͒, and ET͑Gen͒ represent the total energies of the most stable Gen−1, Ge, and Gen clusters, respectively The averaged binding energies and fragmentation energies for the Cu-doped CuGen clusters can be defined as the following formula: EbЈ͑n͒ = ͓ET͑Cu͒ + nET͑Ge͒ − ET͑CuGen͔͒/n + 1, DЈ͑n,n − 1͒ = ET͑CuGen−1͒ + ET͑Ge͒ − ET͑CuGen͒, where ET͑CuGen−1͒, ET͑Ge͒, ET͑Cu͒, and ET͑CuGen͒ represent the total energies of the most stable CuGen−1, Ge, Cu, and CuGen clusters, respectively The calculated results on the averaged binding energies are plotted as the curves which show the sized dependence of the averaged binding energies for the CuGen clusters and Gen clusters As shown in Table II and Fig 4, the averaged binding energy increases dramatically as the size of Gen changes from to 4; furthermore, the averaged binding energy in- Downloaded 10 Jun 2013 to 134.58.49.37 This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions 244303-9 J Chem Phys 123, 244303 ͑2005͒ Copper-doped germanium clusters TABLE II The averaged binding energies and fragmentation energies of the lowest-energy structure Cluster BE ͑eV͒ FE ͑eV͒ Cluster BE ͑eV͒ BEa ͑eV͒ BEb ͑eV͒ FE ͑eV͒ CuGe2 CuGe3 CuGe4 CuGe5 CuGe6 CuGe7 CuGe8 CuGe9 CuGe10 CuGe11 CuGe12 CuGe13 1.76 2.01 2.19 2.39 2.46 2.51 2.54 2.63 2.75 2.72 2.72 2.72 2.76 2.91 3.43 2.84 2.86 2.78 3.4 3.99 2.35 2.73 2.73 Ge2 Ge3 Ge4 Ge5 Ge6 Ge7 Ge8 Ge9 Ge10 Ge11 Ge12 Ge13 1.44 1.98 2.37 2.46 2.55 2.64 2.58 2.66 2.73 2.69 2.68 2.63 1.23 2.24 2.7 2.91 3.05 3.22 3.16 3.24 3.33 3.27 3.26 3.29 1.35 2.04 2.53 2.72 2.85 2.97 3.06 3.04 3.13 3.13 3.21 3.12 3.06 3.52 2.83 2.99 3.17 2.19 3.31 3.37 2.23 2.63 2.1 a Theoretical binding energy per atom using DFT-GGA from Ref 47 Experimental binding energy per atom ͓for Ge2–8, measured atomization energy ͑Refs 23, 48, and 49͒ for Ge9–13, estimation from ion mobility ͑Ref 50͔͒ b creases smoothly when the size of Gen clusters increases from to 10 However, when the size of Gen exceeds 10, the averaged binding energy is gradually decreased as evinced by drop of the stability It reflects that the stabilities of the small-sized Gen͑n ഛ 10͒ clusters have been enhanced with increase of the size of clusters, and that the decrease of averaged binding energy of the large-sized Gen͑n Ͼ 10͒ clusters predicts restriction of the formation of large-sized Gen clusters The prediction about averaged binding energy of the pure Gen clusters in this work is in agreement with the previous calculation results and experimental results.47,23,48–50 When Cu is doped on the pure germanium clusters, the averaged binding energy is obviously influenced by Cu As shown in Fig 4, the averaged binding energy of the CuGen͑n ഛ 10͒ is lower than that of the Gen+1 In contrast, the averaged binding energy of the encapsulated CuGen͑n Ͼ 10͒ clusters is higher than that of identical-sized Gen clusters For example, the averaged binding energy ͑2.718 eV͒ of the encapsulated CuGe13 13bЈ is higher than that ͑2.631 eV͒ of the Ge13 It reveals that the encapsulation of Cu in the large-sized Gen͑n Ͼ 10͒ increases its averaged binding energy and contributes to enhancing the stability of the caged Gen clusters, and that the doped Cu atom elevates the averaged binding energy of the large-sized Gen clusters as compared to the pure Gen clusters As can be seen from Fig 4, the averaged binding energy of the CuGen clusters increases monotonously to maximum as the size of the CuGen clusters increases from to 10 However, the averaged binding energy of the CuGen clusters is gradually decreased as the size of the CuGen exceeds 10 On the other hand, the sized dependence of the fragmentation energies of the Gen and CuGen clusters is also investigated As shown in Fig 5, the fragmentation energy decreases with increase of the size of the small Gen clusters However, it increases when the size of Gen clusters is bigger than Our surprising finding is that the fragmentation energy sharply reaches to maximum at the Ge10 and decreases dramatically after the Ge10 cluster, exhibiting that the Ge10 is the most stable isomer This finding is in good agreement with the experimental measurement on the maximum ioniza- FIG Sized dependence of the atomic binding energies of CuGen and Gen ͑n = – 13͒ clusters FIG Sized dependence of the fragmentation energies of CuGen and Gen ͑n = – 13͒ clusters Downloaded 10 Jun 2013 to 134.58.49.37 This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions 244303-10 J Chem Phys 123, 244303 ͑2005͒ J Wang and J.-G Han tion potential of the Gen clusters being n = 10.25 Subsequently, the local abundant peaks of the D͑n , n − 1͒ appear at the sizes of 7, 9, 10, and 12 However, when Cu is doped in the Gen clusters, this situation is obvious As can be seen from Fig 5, the local maxima of DЈ͑n , n − 1͒ appear at the sizes of and 10 Especially, the fragmentation energy of the CuGe12 cluster is not distinctly higher than that of the CuGe13 cluster It indicates that the relative stability of the CuGe12, which is different from Ta– Si12,44 is stronger than that of the adjacent CuGe11, but is identical to the CuGe13 cluster It should be pointed out that the binding energy and fragmentation energy of the CuGe10 are the highest among all sized clusters, indicating that the relative stability of the CuGe10 cluster is stronger than other sized clusters, which is in good agreement with the experimental observation on the − 51 As observed from the optimized CuGe10 structure, CoGe10 all the Cu–Ge bond lengths and electrostatic interactions among Cu and all the Ge atoms are almost equal As for basketlike CuGe12 12dЈ and CuGe13 13bЈ structures, some of Ge atoms cannot efficiently interact with the Cu atom, which are reflected from some Ge atoms being far from the concaved Cu atom; this finding implies that the number of dangling bonds of the Ge atoms in the basketlike structures is more than that of the bicapped tetragonal antiprism or multirhombus CuGe10 10aЈ Hence, it can be expected that the encapsulated CuGe10 10aЈ cluster can be acted as building block of the Cu-doped clustered-assembled materials because of the fourfold coordination of Ge atoms and no appreciable Ge dangling bonds In addition, for all CuGen ͑n Ͼ 9͒ clusters, the averaged binding energy and fragmentation energy are significantly higher than those of the comparable sized Gen clusters, and reflecting that the stability of CuGen is enhanced when Cu is doped in the Gen clusters Beginning from the CuGe8, each sized Cu-doped cluster has a stable encapsulated structure In order to investigate the stability of encapsulated structure with the variation of size, it is necessary to calculate the embedding energy of the Cudoped Gen frameworks And the embedding energy can be defined as the following formula: FIG Sized dependence of the embedding energies of CuGen ͑n = – 13͒ clusters EE = ET͑Cu͒ + ET͑Gen͒ − ET͑CuGen͒, where ET͑Cu͒, ET͑Gen͒, and ET͑CuGen͒ represent the total energies of the most stable Cu, Gen and CuGen clusters, respectively As seen from Fig 6, the embedding energy increases rapidly when the size of the CuGen changes from to 10, then the embedding energy increases smoothly from 10 to 12 When size of CuGen varies from 12 to 13, the embedding energy increases rapidly again These findings indicate that the extrastability of the encapsulated Cu atom in the small Gen frames is not as strong as that of the encapsulated Cu atom in the large Gen cages with increase of the size of the CuGen clusters E HOMO-LUMO gap, charge transfer, and polarizability Semiconducting characters of semiconductive material germanium and the TM-doped germanium clusters can be reflected from the energy gap between HOMO and LUMO TABLE III The natural charge population, HOMO-LUMO gap, and dipole moment of the lowest-energy structures of different sized CuGen and Gen ͑n = – 13͒ clusters Cluster Natural population HOMO-LUMO gap ͑eV͒ Dipole moment Cluster HOMO-LUMO gap ͑eV͒ Dipole moment CuGe2 CuGe3 CuGe4 CuGe5 CuGe6 CuGe7 CuGe8 CuGe9 CuGe10 CuGe11 CuGe12 CuGe13 0.281 0.386 0.474 0.492 0.461 0.587 0.547 0.354 0.259 0.332 0.446 0.476 1.376 1.741 1.673 1.755 1.521 1.412 1.266 1.219 1.191 1.193 1.235 1.144 1.836 1.214 1.403 1.722 1.805 1.204 1.51 0.296 0.021 0.984 0.216 1.079 Ge2 Ge3 Ge4 Ge5 Ge6 Ge7 Ge8 Ge9 Ge10 Ge11 Ge12 Ge13 0.804 2.642 2.282 2.963 2.782 2.168 2.209 2.647 2.622 2.111 1.705 1.637 0.613 0.001 0.155 0.001 0.543 0.76 0.561 0.893 1.002 1.835 Downloaded 10 Jun 2013 to 134.58.49.37 This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions 244303-11 As seen from Table III, the HOMO-LUMO gaps of CuGen are distinctly lower than those of Gen clusters Previous investigation suggested that charge-transferring direction could be changed when the size of the TM-doped Sin clusters ͑TM= Ni, Ta, and Zr͒ increases.32,44,52 However, according to natural population analysis of the CuGen clusters, charge always transfers from Cu to the Ge atoms with the increase of size, and indicates that Cu acts as electron donor in all CuGen clusters Local maximum of the charge transfer and relative stability for the different sized CuGen clusters can be found at 10; this finding provides a support of our calculated relative stability by aid of the calculated fragmentation energy It can be reflected that the charge transfer in the encapsulated symmetrical CuGen is large and contributes to enhancing proportionate electrostatic interaction which plays an important role in stabilizing the encapsulated structures Wang et al.21 suggested that the dipole moments of Gen ͑n = , , , 7͒ are nearly zero corresponding to high symmetry In our calculation, similar results on the dipole moments of the Gen clusters can be also obtained Additionally, Wang et al also pointed out that polarizability of the Gen clusters increases with decreasing HOMO-LUMO gaps; however, no direct relationship between the polarizability and HOMOLUMO gap can be found for the CuGen clusters For example, as seen from the optimized lowest-energy CuGe10 and CuGe11 structures, the dipole moment of the CuGe10 is zero because of the symmetric distribution of the Ge atoms around Cu in the CuGe10 10a’ while the dipole moment of the CuGe11 is close to due to one Ge atom sticking out from the CuGe11 11a’ structure; however, their HOMOLUMO gaps are almost identical Hence, it can be expected that the polarizabilities of the CuGen clusters are mainly dependent on symmetric distribution of the Ge atoms around Cu The dipole moment of the CuGen isomers generally decreases along with the increase of the dipole moment of the Gen clusters as the size of the Gen and CuGen clusters increases, indicating that the Cu atom is doped in the center sites of the Gen frames as the size of the CuGen clusters increases IV CONCLUSION The growth behaviors, stabilities, electronic properties, and polarizabilities of the Gen and CuGen ͑n = – 13͒ clusters are investigated theoretically at the UB3LYP level employing LanL2DZ basis sets All the calculated results are summarized as follows ͑1͒ J Chem Phys 123, 244303 ͑2005͒ Copper-doped germanium clusters According to optimized geometries of the Gen and CuGen ͑n = – 13͒ clusters, it is found that the growth behaviors of the Cu-doped Gen clusters are different from the pure Gen clusters For the pure Gen clusters, three different growth patterns, i.e., in-plane, out-ofplane edge-capped, or face-capped growth patterns, are dominant for the different sized Gen clusters However, four different growth patterns, i.e., Cu-capped, Cusubstituted, Cu-concaved, and Ge-capped patterns, are dominant for the different sized CuGen clusters ͑2͒ ͑3͒ ͑4͒ According to the averaged binding energy analyses of the Gen and CuGen clusters, it is concluded that the doped Cu in the small-sized Gen ͑n Ͻ 10͒ clusters decreases the binding energies while the doped Cu in large-sized Gen ͑n Ͼ 10͒ clusters increases the binding energies The calculated results on fragmentation energies of the Gen and CuGen clusters indicate that the relative stabilities of the Cu-doped Gen ͑n ജ 10͒ clusters are enhanced as compared to the pure Gen clusters ͑n ജ 10͒ The magic numbers of the stabilities are 7, 9, 10, and 12 for the Gen clusters, and and 10 for the CuGen clusters Furthermore, the relative stabilities of the Cu-doped Gen ͑n = , 10, 13͒ are obviously enhanced as compared to the identical-sized Gen clusters Moreover, the relative stability of the CuGe10 isomer turns out to be the most stable cluster which is con− firmed by the experimental measurement of the CoGe10 48 isomer Although the doped Cu in the Gen clusters does not seriously distort the pure Gen frames, however, the chemical bonding type of the HOMO is distinctly changed as compared to the pure Gen clusters It should be mentioned that the analysis of the HOMO properties contributes to explaining the growth behavior of the CuGen clusters The HOMO-LUMO gap of the CuGen clusters decreases obviously as compared to the pure Gen clusters Unlike TM-Sin ͑TM= Ni, Zr, Ta, etc.͒, the charge in the CuGen clusters always transfers from the Cu atom to the Ge atoms In addition, the relationship between the polarizabilities and HOMO-LUMO gaps for the pure Gen clusters is destroyed in the Cu-doped Gen clusters ACKNOWLEDGMENTS This work is supported 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the abstract Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions [...]... Dipole moment Cluster HOMO-LUMO gap ͑eV͒ Dipole moment CuGe2 CuGe3 CuGe4 CuGe5 CuGe6 CuGe7 CuGe8 CuGe9 CuGe10 CuGe11 CuGe12 CuGe13 0.281 0.386 0.474 0.492 0.461 0.587 0.547 0.354 0.259 0.332 0.446 0.476 1.376 1.741 1.673 1.755 1.521 1.412 1.266 1.219 1.191 1.193 1.235 1.144 1.836 1.214 1.403 1.722 1.805 1.204 1.51 0.296 0.021 0.984 0.216 1.079 Ge2 Ge3 Ge4 Ge5 Ge6 Ge7 Ge8 Ge9 Ge1 0 Ge1 1 Ge1 2 Ge1 3 0.804... energy analyses of the Gen and CuGen clusters, it is concluded that the doped Cu in the small-sized Gen ͑n Ͻ 10͒ clusters decreases the binding energies while the doped Cu in large-sized Gen ͑n Ͼ 10͒ clusters increases the binding energies The calculated results on fragmentation energies of the Gen and CuGen clusters indicate that the relative stabilities of the Cu- doped Gen ͑n ജ 10͒ clusters are enhanced... polarizabilities of the CuGen clusters are mainly dependent on symmetric distribution of the Ge atoms around Cu The dipole moment of the CuGen isomers generally decreases along with the increase of the dipole moment of the Gen clusters as the size of the Gen and CuGen clusters increases, indicating that the Cu atom is doped in the center sites of the Gen frames as the size of the CuGen clusters increases... of size, it is necessary to calculate the embedding energy of the Cudoped Gen frameworks And the embedding energy can be defined as the following formula: FIG 6 Sized dependence of the embedding energies of CuGen ͑n = 8 – 13͒ clusters EE = ET Cu + ET͑Gen͒ − ET͑CuGen͒, where ET Cu , ET͑Gen͒, and ET͑CuGen͒ represent the total energies of the most stable Cu, Gen and CuGen clusters, respectively As seen... polarizabilities of the Gen and CuGen ͑n = 2 – 13͒ clusters are investigated theoretically at the UB3LYP level employing LanL2DZ basis sets All the calculated results are summarized as follows ͑1͒ J Chem Phys 123, 244303 ͑2005͒ Copper -doped germanium clusters According to optimized geometries of the Gen and CuGen ͑n = 2 – 13͒ clusters, it is found that the growth behaviors of the Cu- doped Gen clusters are different... fourfold coordination of Ge atoms and no appreciable Ge dangling bonds In addition, for all CuGen ͑n Ͼ 9͒ clusters, the averaged binding energy and fragmentation energy are significantly higher than those of the comparable sized Gen clusters, and reflecting that the stability of CuGen is enhanced when Cu is doped in the Gen clusters Beginning from the CuGe8, each sized Cu- doped cluster has a stable encapsulated... polarizability and HOMOLUMO gap can be found for the CuGen clusters For example, as seen from the optimized lowest-energy CuGe10 and CuGe11 structures, the dipole moment of the CuGe10 is zero because of the symmetric distribution of the Ge atoms around Cu in the CuGe10 10a’ while the dipole moment of the CuGe11 is close to 1 due to one Ge atom sticking out from the CuGe11 11a’ structure; however, their HOMOLUMO... relative stability of the CuGe10 cluster is stronger than other sized clusters, which is in good agreement with the experimental observation on the − 51 As observed from the optimized CuGe10 structure, CoGe10 all the Cu Ge bond lengths and electrostatic interactions among Cu and all the Ge atoms are almost equal As for basketlike CuGe12 12dЈ and CuGe13 13bЈ structures, some of Ge atoms cannot efficiently... fragmentation energy of the CuGe12 cluster is not distinctly higher than that of the CuGe13 cluster It indicates that the relative stability of the CuGe12, which is different from Ta– Si12,44 is stronger than that of the adjacent CuGe11, but is identical to the CuGe13 cluster It should be pointed out that the binding energy and fragmentation energy of the CuGe10 are the highest among all sized clusters, indicating... http://jcp.aip.org/about/rights_and_permissions 244303- 11 As seen from Table III, the HOMO-LUMO gaps of CuGen are distinctly lower than those of Gen clusters Previous investigation suggested that charge-transferring direction could be changed when the size of the TM -doped Sin clusters ͑TM= Ni, Ta, and Zr͒ increases.32,44,52 However, according to natural population analysis of the CuGen clusters, charge always transfers from Cu to the Ge atoms