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Computational and Theoretical Chemistry 1054 (2015) 8–15 Contents lists available at ScienceDirect Computational and Theoretical Chemistry journal homepage: www.elsevier.com/locate/comptc A computational investigation of aluminum-doped germanium clusters by density functional theory study Shunping Shi a,⇑, Yiliang Liu b, Chuanyu Zhang a, Banglin Deng a, Gang Jiang c a Department of Applied Physics, Chengdu University of Technology, Chengdu 610059, China College of Electrical and Information Engineering, Southwest University for Nationalities, Chengdu 610041, China c Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China b a r t i c l e i n f o Article history: Received September 2014 Received in revised form 27 October 2014 Accepted December 2014 Available online 11 December 2014 Keywords: Density functional theory Gen+1 cluster GenAl clusters Structure of clusters a b s t r a c t We report a computational study of the aluminum doped germanium clusters GenAl (n = 1–9) The molecular geometries and electronic structures of the GenAl clusters are investigated systematically using quantum calculations at the B3LYP level with the 6-311G(d) basis sets The growth pattern behaviors, stabilities, electronic properties, and magnetic moments of these clusters are discussed in detail Obviously different growth patterns appear between small and larger Al-doped germanium clusters, the optimized equilibrium geometries trend to prefer the close-packed configurations for Al-doped germanium clusters up to n = The size dependence of cluster average binding energies per atom (Eb/atom), second-order differences of total energies (D2E), fragmentation energies (Ef) and HOMO–LUMO gaps of Gen+1 and GenAl (n = 1–9) clusters are studied The stability results show that Gen+1 cluster possess relatively higher stability than GenAl cluster Furthermore, the investigated highest occupied molecular orbital-lowest unoccupied molecular orbital gaps indicate that the Gen+1 and GenAl clusters have different HOMO– LUMO gap In addition, the calculated vertical ionization potentials and vertical electron affinities confirm the electric properties of Gen+1 and GenAl clusters Besides, the doping of Al atom also brings the decrease as the cluster sizes increase for atomic magnetic moments (lb) Ó 2014 Elsevier B.V All rights reserved Introduction The semiconductor clusters with transition metal have attracted great interest for optoelectronic materials, catalyst, and the development of new species in nanoscale applications Germanium clusters have also widely been studied because they are important for the fine processing of semiconductors and the synthesis of novel materials The studies have shown that the structure and the bonding of bulk germanium are very similar to that of bulk silicon, and the bulk surfaces show similar reconstruction [1] However, although small silicon and germanium clusters appear to have similar geometries, the larger ones are fundamentally different [2] During the past two decades, Gen clusters have been intensively studied both experimentally [3–9] and theoretically [10–21] because of their fundamental importance and potential applications in nanoelectronics The photoionization study has been investigated by Yoshida and Fuke to characterize the electronic structures of germanium cluster, they found a rapid decrease in the ionization potentials (IPs) for Gen between n = 15 ⇑ Corresponding author Tel.: +86 2884078267; fax: +86 28 85415508 E-mail address: shishunping13@cdut.cn (S Shi) http://dx.doi.org/10.1016/j.comptc.2014.12.004 2210-271X/Ó 2014 Elsevier B.V All rights reserved and 26, which was very similar to that for silicon clusters [4] The low-lying stages of Ge2 and GeÀ have also been probed using negative ion zero electron kinetic energy spectroscopy [7] Because of the lack of experimental method to characterize the structure of germanium clusters, most of the geometrical data come from theoretical calculations Geometrical and electronic properties of Gen (n = 5–10) neutrals, cations, and anions have been investigated using the density functional method of Becke’s three-parameter hybrid functional with the Perdew/Wang 91 expression by Li et al [12] Yoo and Zeng performed a constrained search for the geometries of low-lying neutral germanium clusters in the size range of 21 n 29 [14] Wang et al calculated dipole polarizabilities of Gen clusters at FF level of density functional theory, which show the dipole moment and polarizabilities of Gen clusters are sensitively dependent on the cluster geometries and electronic structures [15] King et al reported the effect of electron count on cluster geometry of nine and ten atom germanium clusters using B3LYP level of DFT [20] The pure germanium clusters are chemically reactive and thus not suitable as a building block of self assembly materials By an appropriate choice of the metal dopant, it is possible to design metallic as well as semiconducting nanotubes using Gen as S Shi et al / Computational and Theoretical Chemistry 1054 (2015) 8–15 building blocks Doped germanium clusters have been performed the focus of a few experimental and theoretical studies [22–35], which exhibit many novel properties such as the sizes selectivity, the highest occupied molecular orbital-lowest unoccupied molecular orbital gap, different charge transfer direction and the magnetic property Tai and Nguyen [22] found the structure and stability of the Ge12Mx clusters with M = Li, Na, Be, Mg, B, Al, and x from À1 to +1, they obtained the high thermodynamic stability of the icosahedra arises from a combination of their closed crystal field shells, spherical aromaticity and electrostatic attraction force Electronic properties of germanium–fluorine cluster anions (GenFmÀ; n = 1–11, m = 1–3) were studied by Negishi et al using photoelectron spectroscopy with a magnetic-bottle type electron spectrometer, which showed that the doped F atom in GenFÀ deprives each GenÀ cluster of the excess electron without any serious rearrangement of the Gen framework [23] In addition, the geometries, stability, and electronic properties of TM-doped germanium clusters (TM = Zn, Fe, Mn, Si, Ni, W, Cr, Cu, Au) [25–35] have also been systematically investigated by using different method The remarkable features of Zn-doped Gen clusters are distinctly different from other TM-Gen clusters, indication that the growth pattern of the TM-Gen clusters depends on the kind of doped TM impurity Although many studies have been taken on pure germanium clusters and doped germanium clusters, to our knowledge, surely systematic and theoretical investigated on aluminum-doped germanium clusters have not been reported so far In this work, an investigation on the structures, stabilities, magnetism, and electronic properties of the Al-doped germanium clusters were calculated using density functional theory by considering a considerable number of structural isomers In order to reveal the effect of the doped Al atom to the germanium clusters, in this paper, we optimize the geometrical structures of GenAl (n = 1–9) clusters by employing DFT approach to find the structural and stability, and combined with pure germanium clusters for comparison by using identical methods and basis sets Computational details The geometry optimizations of the Gen+1 and GenAl (n = 1–9) clusters with spin configurations considered are performed by using density functional theory (DFT) with the B3LYP exchange– correlation potential and 6-311G(d) basis sets The B3LYP method, it is based on the Becke three-parameter exchange functional and the Lee, Yang and Parr correlation functional [36,37] In order to test the reliability of our calculations, some test calculations are carried out on Ge2 and Al2 using B3LYP, B3P86, PBE1PBE, and B3PW91 method with LANL2DZ, Def2-TZVP, and 6-311G(d) basis sets The computed spin multiplicities, bond lengths (Re), vibrational frequencies (xe), and dissociation energies (De) of dimers (Ge2, and Al2) and available experimental and previous theoretical data are summarized in Table Comparing with the experimental data, we can find that the B3LYP method with 6-311G(d) basis sets Table The computed spin multiplicities, bond lengths (Re), vibrational frequencies (xe), and dissociation energies (De) of dimers (Ge2, and Al2) and available experimental and previous theoretical data Molecule Method Ge2 B3LYP B3P86 PBE1PBE B3PW91 Spin Re (Å) xe (cmÀ1) De (eV) LanL2DZ 6-311G(d) Def2-TZVP LanL2DZ 6-311G(d) Def2-TZVP LanL2DZ 6-311G(d) Def2-TZVP LanL2DZ 6-311G(d) Def2-TZVP 3 3 3 3 3 3 3a – 2.528 2.413 2.407 2.517 2.388 2.385 2.514 2.386 2.383 2.520 2.393 2.390 2.548b 2.44e 250.1 276.6 279.0 256.4 288.0 290.1 259.4 292.0 294.1 255.8 286.5 288.4 281c 274f 286 + 5g,h 2.34 2.87 2.93 2.52 3.02 3.08 2.40 2.89 2.97 2.42 2.89 2.96 2.34d 2.65f 2.70 ± 0.07g,h LanL2DZ 6-311G(d) Def2-TZVP LanL2DZ 6-311G(d) Def2-TZVP LanL2DZ 6-311G(d) Def2-TZVP LanL2DZ 6-311G(d) Def2-TZVP 3 3 3 3 3 3 3i 3k 2.855 2.765 2.753 2.834 2.737 2.728 2.834 2.739 2.730 2.839 2.746 2.736 2.7i 2.7k 235.1 252.9 259.6 246.4 266.1 271.4 250.7 268.5 274.8 245.9 264.0 270.0 241.2i 284.2k 1.19 1.29 1.36 1.36 1.47 1.53 1.37 1.48 1.54 1.33 1.44 1.49 1.33j 1.34l Theory Experiment Al2 B3LYP B3P86 PBE1PBE B3PW91 Theory Experiment a b c d e f g h i j k l Ref Ref Ref Ref Ref Ref Ref Ref Ref Ref Ref Ref [32] [33] [34] [35] [6] [7] [8] [9] [39] [40] [41] [42] 10 S Shi et al / Computational and Theoretical Chemistry 1054 (2015) 8–15 is more optimal than others Therefore, the B3LYP/6-311G(d) scheme is reliable and accurate enough for describing the systems involving Ge and Al atoms The B3LYP method or 6-311G(d) basis sets was successfully used for pure Gen clusters [12,17] and for GenLi clusters [22], GenCr clusters [31] In this paper, the conformations of the pure Gen+1 clusters are obtained firstly by reference to the configurations in Refs [13,15,18,19] During the course of choosing initial structures of the GenAl clusters, we have considered possible isomeric structures by placing the Al atom on each possible site of the Gen+1 clusters as well as by substituting one Ge atom by the Al atom from Gen+1 cluster Furthermore, different spin states of GenAl clusters are considered and calculated by using the Gaussian 03W package [38], the optimized results are obtained that the most stable structures of the Al-doped germanium clusters The detailed calculated results and discussions are followed Results and discussions 3.1 Lowest-energy structures Using the computation scheme described in Section 2, we have explored a number of low-lying isomers and determined the lowest-energy of GenAl clusters up to n = The obtained ground state geometries and some low-lying metastable isomers are shown in Fig For proper comparison we have also shown the ground state geometries of pure Gen+1 (n = 2–9) clusters The point group symmetries (PG), the spin multiplicities, the electronic states, geometry property, and relative energy DE (relative to lowestenergy structure) of the most stable and low-lying GenAl (n = 1–9) clusters are summarized in Table For Ge2 dimer, the dissociation energy and vibrational frequency are obtained as 2.87 eV and 276.6 cmÀ1 Our current results are in satisfactory agreement with the experimental data (dissociation energy De = 2.7 eV and x = 286 cmÀ1) [8,9] The Ge–Ge bond length for Ge2 dimer is predicted to be 2.413 Å, this is also consistent with results 2.44 Å of experimental [6] The electronic state and spin multiplicity are 3Rg and triplet spin state, respectively, which agrees well with the results of Deutsch et al [16] and Bandyopadhyay and Sen [32] For the GeAl monomer with C1v symmetries, the optimized results indicate that the quadruple spin state is lowest energy Therefore, the quadruple GeAl monomer with a bond length of 2.491 Å is most stable structure, the corresponding electronic state is 4R For Ge3, the isosceles triangle structure is suggested as the lowest energy structure with an apex angle of 83.8° corresponding to the 1A0 Our result is excellent with previous theory [19], in which the isosceles triangle structure with an apex angle 84.9° The most stable structure of Ge2Al cluster is also an isosceles triangle structure (3-a) This configuration presents the low spin state of 1A0 The Ge–Ge bond length, the Ge–Al bond length, and the vibrational frequencies of ground 1A0 state of Ge2Al cluster is 2.585 Å, 2.408 Å, and 167.4 cmÀ1, respectively The linear C1v (Ge–Ge–Al) configuration is also considered, corresponding to 3-b isomer with 1A00 state, the linear C1v structure is lower in energy than the linear Cs isomers The linear Ge–Al–Ge geometry with quartet spin multiplicity is also found to be stable However, the configuration corresponds to very high relative energies of 1.120 eV The Ge4 is a rhombus structures with C2V symmetry, the corresponding to bond length is 2.475 Å and electronic state is 1A1 Five kinds of Ge3Al clusters can be optimized to the minima When n = 3, the planar structures (3-a) are proved to be the lowestenergy structures, but three-dimensional (3D) structures (3-b, 3-c, 3-d, and 3-e) are not the most stable structures in our calculated clusters The Ge–Ge–Ge bond angle (104.3°) of the 3-a isomer, generated from substitution of Ge4 3-a0 rhombus by Al, is much larger than that of the Ge3 cluster The 3-b isomer is a distorted Y-type structure, which can be described as one Al atom being bonded on the apical Ge atom in the lowest energy Ge3 cluster If one Al or Ge atom is capped on the lowest energy 2-a0 or 2-a, the 3-c or 3-d isomer may be formed 3-e is the highest energy between planar structures and 3D structures The most stable structure (3-a) have CS symmetry and 2A0 electronic state The most favorable geometry of Ge5 cluster is a distorted trigonal bipyramid structure with D3H symmetry, corresponding to the A state The lowest energy structure of Ge4Al is 4-a in C1 symmetry with 2A electronic state, which is formed by capping one Al atom on the top of Ge4 cluster 4-b and 4-c can be viewed as Al atom substituted a Ge atom from the apical and middle in the Ge5 isomer, which are obvious higher in energy than the lowest energy structure 4-a by 0.548 and 0.748 eV, respectively One planar isomer (4-d) with C1 symmetry behaves the highest energy (DE = 1.076 eV) among all isomers of Ge4Al clusters Therefore, from n = 4, the 3D structures are more stable than the planar structures (4-a > 4-b > 4-c > 4-d) The distorted octahedron is obtained for Ge6, which has D3 symmetry, corresponding to the electronic is 3A1 Four structures obtained for Ge5Al clusters have C1 symmetry The most stable (5-a) with 2A electronic state, corresponding to the Ge–Ge bond length, and the Ge–Al bond length are 2.612 Å and 2.802 Å The prism structure (5-b) and the distorted octahedron structure (5-c) usually are considered the most stable, but in our calculation, their energies higher than the 5-a, which are 0.441 and 0.603 eV, respectively The structure of 5-d isomer with C1 symmetry and spin multiplicity (PG = 2), which relation energy is 1.144 eV In the case of n = 7, the pure Ge7 adopted the pentagonal bipyramid structure with C1 symmetry, corresponding to the electronic state and Ge–Ge bond length are 1A and 2.681 Å, respectively Although geometries structure of Ge7 cluster is same as Refs [19,32], the symmetry is different Four different isomers are found for Ge6Al The most stable structure (6-a) is obtained by one Al atom substitute one Ge atom from the waist of pure Ge7 cluster (6-a0), which has C1 symmetry, and its the Ge–Ge bond length, and the Ge–Al bond length are 2.697 Å and 2.636 Å The other pentagonal bipyramid structure (6-b) is one Al atom substitute one Ge atom from the top of pure Ge7 cluster (6-a0) The relation energy of 6-c and 6-d are 0.607 and 0.827 eV, respectively When the size of Gen clusters is up to 8, one structure, which is obtained from the pentagonal bipyramid Ge7 (6-a0), is proven to be stable structure The Ge–Ge bond length, and the vibrational frequencies of ground 1A state of Ge8 cluster are 2.681 Å, and 86.5 cmÀ1, respectively The most stable Ge7Al (7-a) cluster can be generated from one Al atom substitute one Ge atom on the lowest energy Ge8 cluster It displays C1 symmetry with 2A electronic state The 7-b, 7-c, and 7-d isomers have higher energies compared with the 7-a structure in its ground state by 0.051, 0.249, and 0.641 eV, respectively As for Ge9, the lowest-energy structure is a bicapped pentagonal bipyramid structure with C1 symmetry, corresponding to the spin multiplicity is The configuration of Ge9 can be easily understood as growth on the basis of Ge8 Five kinds of stable structures can be verified to be the minima in Ge8Al The most stable structure 8-a with C1 symmetry, it is seen that the Al atom substitutes one Ge atom of the Ge9 cluster Other possible isomers (8-b–8-e) have energies higher than 8-a by 0.202, 0.260, 0.276, 0.905 eV Especially, the 8-e isomer is much higher in energy than 8-a by 0.905 eV Although both Ge8Al 8-c and 8-d isomers are close in energy, they have different structures The ground-state structure obtained for Ge10 has C1 symmetry and it can be built from Ge9 wedges The electronic state and the Ge–Ge bond length are 1A and 2.728 Å, respectively For Ge9Al, 11 S Shi et al / Computational and Theoretical Chemistry 1054 (2015) 8–15 1-a0 1-a 2-a0 2-a 2-b 2-c 3-a0 3-a 3-b 3-c 3-d 3-e 4-a0 4-a 4-b 4-c 4-d 5-a0 5-a 5-b 5-c 5-d 6-a0 6-a 6-b 6-c 6-d 7-a0 7-a 7-b 7-c 7-d 8-a0 8-a 8-b 8-c 8-d 8-e 9-a0 9-b 9-c 9-a 9-d Fig Ground-state configurations and low-lying isomers of GenAl (n = 1–9) clusters and the lowest-energy structures of pure Gen+1 (n = 1–9) clusters The first GenAl structure is the lowest-energy one for GenAl (n = 2–9) the four most stable isomers are listed in Fig (9-a–9-d), although from 9-a to 9-d isomers are different in structure, they have same symmetry (C1) and electronic state (2A) The lowest energy isomer is in C1 symmetry, a multirhombus prism with one side capped Ge 12 S Shi et al / Computational and Theoretical Chemistry 1054 (2015) 8–15 Table The point group symmetries (PG), spin multiplicity, electronic state, bond lengths and vibrational frequencies, and relative energies of Gen+1 and GenAl (1–9) clusters Rav1 and Rav2 denote the average bond lengths of Ge–Ge and Ge–Al, respectively; Freq denotes the lowest vibrational frequency of the Gen+1 and GenAl equilibrium geometry Cluster Ge2 GeAl Ge3 Ge2Al Ge4 Ge3Al Ge5 Ge4Al Ge6 Ge5Al Ge7 Ge6Al Ge8 Ge7Al Ge9 Ge8Al Ge10 Ge9Al Isomer 1-a0 1-a 2-a0 2-a 2-b 2-c 3-a0 3-a 3-b 3-c 3-d 3-e 4-a0 4-a 4-b 4-c 4-d 5-a0 5-a 5-b 5-c 5-d 6-a0 6-a 6-b 6-c 6-d 7-a0 7-a 7-b 7-c 7-d 8-a0 8-a 8-b 8-c 8-d 8-e 9-a0 9-a 9-b 9-c 9-d PG D1h C1v Cs Cs C1v C2v C2v Cs C1 C3v Cs Cs D3H C1 C2v C3v C1 D3 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 CS C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 Spin State Rav1 (Å) Rav2 (Å) Freq (cmÀ1) DE (eV) 2 4 6 4 2 2 2 2 2 2 2 2 2 2 2.403 – 2.842 2.585 2.306 5.006 2.475 2.424 2.505 2.664 2.664 2.564 2.645 2.520 2.603 2.682 2.505 2.655 2.612 2.590 2.619 2.611 2.681 2.697 2.693 2.640 2.702 2.681 2.722 2.810 2.671 2.692 2.714 2.736 2.619 2.785 2.706 2.770 2.728 2.729 2.757 2.786 2.737 – 2.491 – 2.408 2.598 2.503 – 2.491 2.573 2.699 2.700 2.817 – 2.919 2.752 2.695 2.594 – 2.802 2.554 2.711 2.572 – 2.636 2.722 2.711 2.690 – 3.323 3.247 2.987 2.581 – 2.625 2.697 2.973 2.998 2.778 – 2.687 2.583 2.590 2.601 276.6 293.0 94.7 167.4 337.7 49.6 53.7 15.3 7.8 135.4 135.4 90.2 38.5 81.2 72.9 54.3 14.6 58.2 20.2 33.3 16.5 24.4 86.5 78.2 75.4 9.5 30.7 86.5 27.8 38.3 42.0 54.5 48.3 46.4 45.7 42.5 34.7 28.1 6.2 34.2 36.9 45.8 31.5 0.00 0.00 0.00 0.00 0.143 1.120 0.00 0.00 1.492 1.624 1.625 2.380 0.00 0.00 0.548 0.748 1.076 0.00 0.00 0.441 0.603 1.144 0.00 0.00 0.040 0.607 0.827 0.00 0.00 0.051 0.249 0.641 0.00 0.00 0.202 0.260 0.276 0.905 0.00 0.00 0.276 0.361 1.043 Rg R A0 A0 00 A B2 A1 A A A1 A A A A A2 A1 A A1 A A A A A A A A A A A A A A A A A A A A A A A A A atom and the other side capped Al atom, which can be view as one Ge atom capped on the 8-d cluster Other possible isomers (8-b–8d) have energies higher than 8-a by 0.276, 0.361, and 1.043 eV as shown in Table 3.2 Relative stability of different sized GenAl clusters The understanding of the relative stability of different sized GenAl (1–9) clusters is important for novel cluster-assembled optoelectronic materials and can provide a good way to show the relative local stability of small clusters So the relative stability of different GenAl clusters can be represented with the average binding energies (Eb), second-order differences of total energies (D2E) and fragmentation energies (Ef) Firstly, we consider the corresponding Eb, D2E, and Ef of Gen+1 (n = 1–9) clusters to provide an interpretation They are expressed as Eb ½Genþ1 Š ¼ ððn þ 1ÞE½GeŠ À E½Genþ1 ŠÞ=ðn þ 1Þ ð1Þ D2 E½Genþ1 Š ¼ E½Genþ2 Š þ E½Gen Š À 2E½Genþ1 Š Ef ½Genþ1 Š ¼ E½Gen Š þ E½GeŠ À E½Genþ1 Š ð2Þ ð3Þ where E[Ge], E[Gen], E[Gen+1], and E[Gen+2]denote the total energies of the lowest energy Ge, Gen, Gen+1, and Gen+2 clusters, respectively For GenAl (n = 1–9) clusters, the average binding energies (Eb), second-order differences of total energies (D2E) and fragmentation energies (Ef) can be calculated by following formulas: Eb ½Gen AlŠ ¼ ðnE½GeŠ þ E½AlŠ À E½Gen AlŠÞ=ðn þ 1Þ ð4Þ D2 E½Gen AlŠ ¼ E½Genþ1 AlŠ þ E½GenÀ1 AlŠ À 2E½Gen AlŠ Ef ðGen AlÞ ¼ EðGenÀ1 AlÞ þ EðGeÞ À EðGen AlÞ ð6Þ ð5Þ where E[Ge], E[Al], E[GenÀ1Al], E[GenAl], and E[Gen+1Al], respectively, are the total energies of the stable atoms or clusters for Ge, Al, GenÀ1Al, GenAl, and Gen+1Al Based on the above formulas, the calculated results of average binding energies, second-order differences of total energies and fragmentation energies are shown in Figs 2–4 From Fig 2, in general, it can be seen that binding energies of Gen+1 clusters increase with cluster size up to n = and contain one minor bump at n = implying that the cluster for n = is more stable than their neighbors, our results are excellent with previous theory [19,33] When Al is doped on the pure germanium clusters, the averaged binding energy increase smoothly with the size of GenAl clusters increases from to 9, the tendency is almost consistent with the binding energy of Gen+1 cluster Although, the average binding energy increase unceasingly, the average binding energy’s increment speed slows down gradually for n = 1–3, 3–5, 5–7, 7–9 In general, the average binding energy grows gradually as clusters size n and 13 S Shi et al / Computational and Theoretical Chemistry 1054 (2015) 8–15 Binding Energy per atom (eV) 3.0 2.8 2.6 2.4 2.2 Gen+1 GenAl 2.0 1.8 1.6 1.4 1.2 1.0 10 Cluster size n Fig Calculated binding energy per atom of germanium clusters and Al-doped Ge clusters (n = 1–9) plotted as function of number of Ge atoms The Second-order Differences (eV) Gen+1 GenAl 3.3 Homo–Lumo gap -1 -2 Cluster size z Fig Calculated second difference in energies for the germanium clusters and Al-doped Ge clusters as a function of number of Ge atoms 4.5 Gen+1 3.5 Gen+1 GenAl 4.0 The calculations of molecular orbital were based on the Hückel method proposed by Erich Hückel in 1930, which is a linear combination of atomic orbital (LCAO) method If the closed electronic configuration of a cluster has a large highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) gaps, which show the cluster contain high chemical stability As seen from Fig 5, the Ge2 and Ge6 clusters have a larger HOMO– LUMO gap while the Ge5 and Ge8 clusters have a smaller HOMO– LUMO gap However, when Al is doped on the pure germanium clusters, the GeAl, Ge4Al and Ge8Al clusters have a larger HOMO– LUMO gap while the Ge5Al, Ge7Al and Ge9Al clusters have a smaller HOMO–LUMO gap As shown in Fig 5, for GeAl, Ge4Al and Ge7Al clusters, the HOMO–LUMO gaps of GenAl clusters are usually larger than those of Gen+1 clusters, while for the other clusters, the HOMO–LUMO gaps of GenAl clusters are usually smaller than those Gen Al HOMO-LUMO gap (eV) Fragmentation Energies (eV) present local maximum value at n = 1, 3, and 7, implying that the clusters are more stable than their neighbors However, as shown in Fig 2, the averaged binding energy of GenAl cluster is lower than that of the Gen+1 cluster Fig shows the second-order difference in cluster total energies, as a function of the cluster size The Gen+1 clusters stabilities exhibit pronounced odd–even alternations, but the phenomenon is changed when n = 4, so maxima are found at n = 1, 3, 6, 8, indicating these clusters possess relatively higher stability For GenAl cluster, it is found that the second-order difference in cluster total energies exhibits odd–even oscillations from to Odd n gives high stability while even n gives low stability, but the stability of Ge6Al, Ge7Al and Ge8Al cluster are inverse A distinct characteristic for the stability of the GenAl clusters can also be observed, around n = 6, this behavior is exceptional As shown in Fig 4, it is clearly found that the doping impurity Al atom makes the thermodynamic stability pattern of the host germanium cluster same from n = to n = 9, except for n = And the Gen+1 cluster with higher fragmentation energy than the GenAl clusters, suggesting Gen+1 cluster have higher stability, except for Ge4, Ge5, and Ge7 clusters The local maxima of the fragmentation energy of Gen+1 and GenAl clusters appear at 3, 6, and 3, 5, That is to say the Ge4, Ge7, Ge10 and Ge3Al, Ge5Al, Ge9Al clusters have higher fragmentation energy, indicating that these clusters are more stable than their neighboring ones 3.5 3.0 2.5 2.0 3.0 2.5 2.0 1.5 1.0 1.5 10 Cluster size n Fig Calculated fragmentation energies for the germanium clusters and Al-doped Ge clusters as a function of number of Ge atoms 10 Cluster size n Fig Calculated HOMO–LUMO gaps for the germanium clusters and Al-doped Ge clusters as a function of number of Ge atoms 14 S Shi et al / Computational and Theoretical Chemistry 1054 (2015) 8–15 of Gen+1 clusters From Fig 5, maxima of GenAl cluster is found at n = 1, indication GeAl cluster possesses the highest stability 1.6 1.4 Gen+1 3.4 Electronic properties 1.2 Gen Al In cluster science, electron affinity (EA) and ionization potential (IP) are used as important properties to study the change in electronic structure of the cluster size Vertical electron affinity (VEA) is defined as the energy difference between the anionic and neutral clusters with both at the optimized geometry of the anionic cluster, while the vertical ionization potential (VIP) is defined as the energy difference between the cationic and neutral clusters with both at the optimized geometry of the neutral cluster Chemical hardness (g) is expressed as g = VIP–VEA based on the basis of a finitedifference approximation and the Koopmans theorem [43], and is established as an electronic quantity which may be applied in characterized the relative stability of molecules and aggregate through the principle of maximum hardness (PMH) proposed by Pearson [44] The VIP, VEA and g of the most stable Gen+1 and GenAl (n = 1–9) clusters are calculated and listed in Table The VIP shows an oscillating behavior from Ge2 to Ge10, except for Ge9, the EVA shows an increase from Ge2 to Ge6 and an oscillating behavior from Ge6 to Ge10, therefore, the Ge3 has the maximum hardness while Ge6 has the minimum hardness When Al atom is doped Gen+1 cluster, it clearly sees that doping with Al atom reduces the vertical ionization potential and chemical hardness of germanium clusters, but doped Al atom raise the vertical electron affinity of germanium clusters For GenAl clusters, The VIP shows a decrease from Ge2Al to Ge9Al, except for Ge8Al, while VEA increase with n In addition, the chemical hardness decreases with cluster size 1.0 3.5 Magnetisms Finally, we comment on the magnetic properties of the Gen+1 and GenAl clusters In Fig 6, we compare the magnetic moments of the Gen+1 clusters with computed values for size of the GenAl clusters It is interesting that the atomic averaged magnetic moments are clearly different between the Gen+1 clusters and the GenAl clusters, indicating that the magnetic moments for the GenAl clusters are relatively more larger than the Gen+1 clusters, except Ge6 cluster In Gen+1 system, it should be mentioned that the most stable Ge2 and Ge6 clusters exhibit magnetic moments, but Ge3, Ge4, Ge5, Ge7, Ge8, Ge9, and Ge10 clusters exhibit nonmagnetic ground state, it means that 1-a0 and 5-a0 isomers are the magnetic structures, the other isomers are nonmagnetic structures However, when Al atom is doped germanium clusters, the magnetic moment changes discontinuously with the cluster size and the magnetic moments are decreased with increasing cluster size For the GeAl dimmer, it is 1.50 lb, it decreases up to 0.1 lb for n = Table Vertical ionization potential, vertical electron affinity, chemical hardness of the most stable Gen+1 and GenAl (n = 1–9) clusters (eV) Cluster size n=1 n=2 n=3 n=4 n=5 n=6 n=7 n=8 n=9 Gen+1 GenAl VIP VEA g VIP VEA g 7.627 7.967 7.819 7.956 6.888 7.797 6.875 7.131 7.434 1.751 1.702 1.860 2.125 2.615 1.764 1.956 1.482 1.644 5.876 6.265 5.959 5.831 4.273 6.033 4.919 5.649 5.790 7.599 7.384 7.295 7.444 7.270 7.120 6.239 7.224 7.085 1.139 2.033 1.898 2.531 2.265 2.204 2.108 2.876 3.014 6.460 5.351 5.395 4.913 5.005 4.916 4.131 4.348 4.071 0.8 0.6 0.4 0.2 0.0 10 Cluster size n Fig Size dependence of the average atomic magnetic moments of the lowestenergy Gen+1 and GenAl clusters There is sharply decrease from n = (1.50 lb/atom) to n = (0.33 lb/atom) in the curve of magnetic moments, staring from n = 2, the moments gradually decrease as the cluster sizes increase Conclusions In conclusion, we report a systematic study of the geometric structures, relative stabilities, electronic properties and magnetic properties of Gen+1 and GenAl (n = 1–9) clusters using density functional theory under the generalized gradient approximation scheme Extensive structures and different possible spin states are carefully investigated In order to show the properties of Al atom doped germanium clusters, we also calculate the properties of pure Gen+1 clusters The results can be summarized as follows: (1) According to the optimized equilibrium geometries of the Gen+1 and GenAl clusters, the growth pattern of the Gen+1 and GenAl clusters is investigated Theoretical results indicate that the low-lying isomers for the Gen+1 and GenAl clusters become three dimensional structures when the size n = On the whole, the adopted lowest energy structures of the GenAl are similar to lowest energy structures of the Gen+1 clusters (2) The stability analysis in relation to the calculation of the averaged atomic binding energy, the fragmentation energy, and the second order difference of energy shows that Gen+1 cluster possess relatively higher stability than GenAl cluster The average binding energies of the most stable Gen+1 clusters are higher than those of the GenAl clusters According to the fragmentation energy and the second order difference of energy analysis, it is concluded that the small Ge7 and Ge5Al isomers are the most stable geometries for Gen+1 and GenAl clusters, respectively (3) The HOMO–LUMO gaps are extensively analyzed for Gen+1 and GenAl clusters The obtained results reveal that the GeAl, Ge4Al and Ge8Al clusters have a larger HOMO–LUMO gap while the Ge5Al, Ge7Al and Ge8Al clusters have a smaller HOMO–LUMO gap The VIP of the Gen+1 clusters show an oscillating behavior from Ge2 to Ge10, except for Ge9, the EVA of the Gen+1 clusters show an increase from Ge2 to Ge6 and an oscillating behavior from Ge6 to Ge10 The VIP of the GenAl clusters shows a decrease from Ge2Al to Ge9Al, except for Ge8Al, while VEA of the GenAl clusters 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