Theoretical Study of the SinMgm Clusters and Their Cations: Toward

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Theoretical Study of the SiMg Clusters and Their Cations: Toward Silicon Nanowires with Magnesium Linkers Nguyen Minh Tam, and Minh Tho Nguyen J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b06051 • Publication Date (Web): 28 Jun 2016 Downloaded from http://pubs.acs.org on July 2, 2016

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The Journal of Physical Chemistry

Theoretical Study of the SinMgm Clusters and Their Cations: Toward Silicon Nanowires with Magnesium Linkers

Nguyen Minh Tama,b,* and Minh Tho Nguyenc,* a

Research Group of Computational Chemistry, Ton Duc Thang University, Ho Chi Minh City, Vietnam

b

Faculty of Applied Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam

c

Department of Chemistry, KU Leuven, Celestijnenlaan 200F, B-3001 Leuven, Belgium

*)

Emails: [email protected]; [email protected], Tel: +32-16-327361

(Abstract) We investigated the structures of the singly and doubly magnesium doped silicon clusters in both neutral and cationic states SinMgm0/+ with n = 1 - 10 and m = 1 - 2. Total atomization energies (TAE), heats of formation (ΔHf) and binding energies (BE) were determined using the composite G4 method. BEs of the Mg-doped clusters are decreased with respect to the pure Si counterparts, irrespective of the charge state. As no experimental values are actually available for these systems, the predicted thermochemical values can be used with an expected error margin of ± 3 kcal/mol (± 0.15 eV or ± 12 kJ/mol), due to the uncertainty on the experimental heat of formation of the silicon atom and of the method used. The growth sequence of the singly doped neutral SinMg is similar to that of the singly doped neutral SinLi clusters. In SinMg structures, the Mg atom tends to favor addition on either an edge or a face of the anionic ground state structure Sin- framework. Only in Si8Mg, Mg substitutes a Si atom in the Si9 framework. For the cations SinMg+, the behavior of Mg differs from that of Li. The Mg atom seems to cap on one of edge or face of the cationic Sin+ instead of the neutral bare Sin like as in the case of Li. The doubly Mg1 ACS Paragon Plus Environment


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doped neutral SinMg2 clusters grow up basically following a way comparable to the doubly doped neutral SinX2 with X = Li, Al reported in previous studies. Their growth pattern is that one Mg atom substitutes into a position of Sin+1, whereas the other Mg atom is usually added on an edge, or a face, of the existing cluster. There are however a few exceptions of this observation such as Si10Mg2. In this size, the cyclic framework Si5-Mg-Si5-Mg turns out to be more stable than the cage-type Si10-Mg2. The SinMg2+ cations contain the cationic Sin+1+ frameworks in which one Mg atom actually substitutes into a Si position whereas the remaining Mg atom caps on an edge, or a face. Again, Si10Mg2+ appears as an exception to this trend. The most interesting result of this study is that the Mg dopant, due to its large electron transfer capacity, behaves as a cation Mgδ+ and thereby induces an ionic entity with the Sinδ- anionic partner. The resulting Mg cation can be served as a linker between Sikδ- blocks leading to stabilized linear and cyclic [(Sik)Mg]l structures. In the systems with k =3, 5, 7, 8 and 10, the linear frameworks can be regarded as possible starting blocks for silicon assemblies giving rise to potential 1D-nanowire materials.

1. Introduction In the past several decades, silicon has extensively been used in the semiconductor and optoelectronic industries. In these widespread applications that profoundly affect our daily life, silicon is often used in its bulk solid state. However, bulk silicon can no longer satisfy the current needs of miniaturization of electronic devices. In the intensive search for a device of the future, silicon clusters have emerged as a potential alternative. Such a perspective has been stimulating a wealth of studies of clusters in general, and silicon clusters in particular, as they opened up new avenues for development of nanoscaled materials.1,2

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Defined as entities built up from a few to hundreds of atoms, clusters are intermediates between free atoms and bulk materials. In terms of dimension, the cluster size is in fact reaching the nanoscale range. However, clusters cannot just be considered as small solid particles, since their properties are size-dependent in a nontrivial and nonscalable way. Clusters with enhanced stability behave as superatoms that can further be considered as building blocks for new materials. Therefore, much effort has been invested in the search for clusters having interesting properties and can be tailored for self-assemblies. The presence of a dopant atom usually alters the electronic structure of the cluster, making it possible to modulate the optical response. Of the nanomaterials, nanowires are of particular interest. Attempts to make silicon nanowires as assemblies of small silicon clusters have thus been reported.3,4,5 Previous studies showed that when a Si nanowire is interacting with one Al atom, its electrical conductivity has been found to be enhanced as compared to pristine Si-nanowire.6 Kotlyar et al.7 reported that the Al atoms form an ordered array of magic clusters on the surfaces of Si(111). Paulose and coworkers8 also reported a persistent formation of Al-Si nanowires. Relatively less is known about Si nanowires using other elements as linkers. Recently we carried out several combined experimental spectrometric and theoretical studies on doped silicon clusters SinMm with M = Li, Be, B, Al, C, V, Cu, Mn, Co…9,10,11,12,13,14,15,16,17,18,19,20 at different charge states. Our extensive findings indicated that while the Li and Cu dopants prefer to be absorbed on the surface of silicon hosts, V-doped clusters SinV+ are built up by substituting one Si-atom of the Sin+1+ frameworks by the V-dopant. In addition, endohedrally doped structures with encapsulated impurities were also found at some special sizes. For instance, Be and B were found to be located at the center of the Si8 and Si10 hosts, respectively.10,12 Some transition metals are also encapsulated into larger Sin cages with n = 10, 11, 12, 14 and 16.17,18,21,22,23 We found that the Li and Mn elements emerge as possible



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linkers connecting Si blocks. While Li linkers give rise to low-spin oligomers,24 Mn turns out to be a magnetic copy of silicon.17 Motivated by a search for other elements that could be served as potential linkers in Si nanowires, we set out to perform a systematic investigation on a series of small singly and doubly magnesium doped silicon clusters SinMgm with n = 1-10 and m = 1-2, in both neutral and cationic states. Using the reliable composite G4 method, we determine the geometries of the lowest-lying equilibrium structures and thereby to probe their growth pattern. The G4 energies have effectively been used in our recent studies on boron doped silicon clusters SinB and SinAlm12,13 It turns out that the magnesium element behaves as good linkers for silicon blocks giving rise to a variety of nanowires having different shapes. The present article is organized in the following way. A brief overview of the applied theoretical methodologies will be given. A systematic analysis of the computational results of Mg-doped Si clusters SinMgm examined with n = 1-10 and m = 1-2 will be described. The structural and stability patterns allow us to identify some suitable members that can further be used as superatoms for assemblies. We thus probe the five-atom, seven-atom, eight-atom, and ten-atom Si building blocks, and the role of Mg ions as their linkers. 2. Computational Methods All electronic structure calculations are carried out using the Gaussian 0925 program packages. The searches for energy minima are conducted using different approaches. In the first, we use a stochastic genetic algorithm to generate most possible structures.26 The equilibrium structures that are initially detected using low-level computations, are reoptimized using a higher level method. In the second approach, initial structures of clusters SinMg are manually constructed by either substituting one Si-atom of the Sin+1 frameworks by one Mg-atom, or adding one Mg-atom at various positions on surfaces of the Sin clusters. Similarly, the initial


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structures of SinMg2 clusters are also generated from known structures of pure silicon clusters and singly doped silicon clusters SinMg. In addition, the local energy minima of SinXm clusters previously reported are also used as referenced structures. The use of genetic algorithms is less effective for producing singly doped-clusters having small sizes because most of the relevant structures in silicon clusters are relatively well known. On the contrary, the multiply doped and larger size clusters imply a huge number of initial structures and thus make the genetic search necessary and more effective, even though such a search is quite computationally demanding. However, only a combination of different search approaches allows a consistent set of lower-energy structures to be obtained. While geometry optimizations using low-level computations on initial guess structures are carried out using the hybrid B3LYP functional in conjunction with the 6-31G basis set, all selected equilibrium geometries of SinMgm (n = 1-10, m = 1-2) are fully optimized using the same functional but with the larger 6-311+G(d) basis set.27,28,29 In order to confirm the identity of true local minima obtained, their harmonic vibrational frequencies are also calculated at the same level. Standard enthalpies of formation of the global minima are subsequently evaluated from the corresponding total atomization energies (TAE).30 For this parameter, two sets of calculations are performed using the composite G4 approach.31 In this approach, geometry optimizations and calculations of harmonic vibrational frequencies of clusters are carried out at the B3LYP/631G(2df,p) level, followed by single point electronic energy calculations using the coupledcluster theory CCSD(T) and different additivity corrections to approach the complete basis set (CBS). This composite method has effectively been used to predict the thermochemical properties of a series of small boron clusters32,33,34 and boron-doped silicon clusters.12



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In a recent study, we calibrated the calculated G4 TAE and heats of formation (ΔHf) of a series of small pure silicon clusters (neutral, cationic and anionic Sin, with n = 2 to 13) with respect to the coupled-cluster theory CCSD(T)/CBS approach whose coupled-cluster energies were extrapolated to the complete basis set limit, and also with respect to available experimental values.35 This study pointed out that the G4 heats of formation deviate by an average of ± 2 kcal/mol with respect to the CCSD(T)/CBS counterparts. The available experimental TAE values for pure Si clusters were not quite reliable due to the very large error margins.29 By combining our computed TAE (ΣD0) values determined from G4 calculations, with the known experimental heats of formation at 0 K for the Mg and Si elements, we can derive ΔHf° values at 0 K for the clusters in the gas phase. In this work, we use the values at 0 K ΔHf°(Mg) = 34.87 ± 0.2 kcal/mol, and ΔHf°(Si) = 107.2 ± 0.2 kcal/mol.36 We subsequently obtain the heats of formation at 298 K by following the classical thermochemical procedure.37 Calculated heats of formation at 0 K are then used to evaluate the adiabatic ionization energies (IE) and other energetic quantities. 3. Results and Discussion The shapes of equilibrium structures of the neutral and cationic SinMgm0/+ clusters detected, their symmetry point groups and G4 relative energies are shown in Figures 1, 2, 3 and 4. The G4 total energies of the lowest-lying isomers are listed in Table S1 of the Supporting Information (ESI) file. The calculated values for their heats of formation are given in Table 1. Computed adiabatic ionization energies (IEs) of SinMgm clusters are given in Table 2, and their average binding energies (Eb) tabulated in Table 3. Due to the large number of isomers located on the potential energy surface of each of the clusters considered, we only present in this text some lower-lying isomers whose relative energies are close to the corresponding ground state structure, within a range of ~1.0 eV.


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As for a convention, each structure described hereafter is defined by the label x.n.m.Y in which x = n and c stands for a neutral and cationic state, respectively, n the size of Sin, m the size of Mgm, and finally Y = A, B, C… refers to the different isomers with increasing relative energy ordering. 3.1 Lower-lying Isomers of SinMgm Clusters in Both Neutral and Cationic States In the following section, we briefly describe the main characteristics of the singly and doubly Mg-doped clusters, in terms of their geometry, symmetry point group, spin state and relative energy. Concerning the energy ordering within a cluster system, the structure labeled with the letter A (x.n.m.A) invariably refers to the lowest-lying isomer obtained from G4 calculations. a) The Singly Magnesium-doped SinMg0/+. The main structures are displayed in Figures 1-2. There is a good agreement between our present predictions and previous studies38 on the identity of the global minima of small-sized neutral SinMg with n = 1 – 10 (Figures 1 and 2), except for two sizes. We find that the diatomic neutral SiMg has a high spin state n.1.1.A (3∑-). For the neutral Si8Mg, Fan et al.38 reported that the isomer n.8.1.D (C2v, 1A1) is the most stable structure whereas our G4 results indicate that the two energetically degenerate lowest-lying isomers are n.8.1.A (Cs, 1A’) and n.8.1.B (C1, 1A), both are 0.30 eV lower in energy than n.8.1.D. Both A and B isomers are formed by substituting one of Si atoms of the bicapped pentagonal bipyramid Si9 framework.39 The characteristics of the other sizes have been abundantly discussed in previous studies38,39 and thus do not warrant further comment. Regarding the cationic SinMg+ clusters, their main characteristics can be summarized as follow: i) The doublet isomer c.1.1.A (2Π) is confirmed as the ground state of the cationic dimer SiMg+.



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ii) The global minima of cations Si2Mg+ and Si3Mg+ are the isosceles triangle c.2.1.A (C2v, 2A1) and the planar quadrilateral c.3.1.A (Cs, 2A’), respectively. iii) For Si4Mg+, three isomers c.4.1.A (C2v, 2A1), c.4.1.B (Cs, 2A’), c.4.1.C (C1, 2A) that are generated by capping Mg+ at different positions on the rhombus Si4 framework are the lowestlying structures. The planar structure c.4.1.A is 0.10 and 0.12 eV lower in energy than c.4.1.B and c.4.1.C, respectively. iv) The c.5.1.A (C2v, 2B1) of Si5Mg+ has a three-dimensional shape of the corresponding neutral n.5.1.A in which the Mg-atom caps on an edge of the trigonal bipyramid Si5.39 In other words, Si5Mg in both neutral n.5.1.A and cationic c.5.1.A are formed upon substitution of one Mg atom at a Si position of the Si6 framework which possesses two almost isoenergetic C2v and Cs structures.40 v) For Si6Mg+, our G4 calculations indicate that both lowest-lying isomers c.6.1.A (C2v, 2B2) and c.6.1.B (Cs, 2A’), formed by capping a Mg-dopant on the edge-capped trigonal bipyramid Si6 framework,39 are almost degenerate with energy gap of only 0.09 eV. The neutral Si6Mg n.6.1.A and its corresponding cation c.6.1.A are equally created by replacing one Si atom of the pentagonal bipyramid Si7 39 by a Mg atom. vi) Similarly, the Si7Mg+ c.7.1.A (C2v, 2A1) is generated by adding the Mg-atom on the edge of pentagonal bipyramid Si7 host. vii) For Si8Mg+, c.8.1.A (Cs, 2A’) which is formed by capping the Mg-impurity on the surfaces of Si8+ framework,41,42 is the lowest-lying isomer with energy gap of 0.18 eV below the next isomer, c.8.1.B (C1, 2A), that has the shape of the corresponding neutral n.8.1.A. viii) The Si9Mg+ c.9.1.A (C1, 2A) has the same shape as the ground state of the neutral Si9Mg n.9.1.A in which the Mg-atom is exposed on the surface of the bicapped pentagonal bipyramid Si9+ skeleton.


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ix) For Si10Mg+, most of lower-lying isomers have the tetra-capped trigonal prism framework of Si1039 with the Mg-atom exohedrally adds on different positions. According to our G4 calculations, c.10.1.A (Cs, 2A’) is ~0.3 eV lower in energy than the remaining stable cationic Si10Mg+ structures. b) The Doubly Magnesium-doped SinMg20/+ with n = 1 – 10. Their structural evolution is illustrated in Figure 3 for the sizes n = 1-7 and Figure 4 for the sizes n = 8-11. As these doubly doped clusters have not been reported before, let us briefly describe their main geometrical and electronic characteristics. SiMg2. The triangular isomer n.1.2.A (C2v,

1

A1), possessing a closed-shell electronic

configuration in which the Si-atom connects with two Mg-atoms, is confirmed as the ground state of the neutral SiMg2 with a narrow singlet-triplet separation gap of 0.12 eV. Following detachment of one electron, the resulting cationic cluster SiMg2+ c.1.2.A (C2v, 2B1) retains the shape of its neutral counterpart almost unchanged. Si2Mg2. All lowest-lying structures in both neutral and cationic states have planar geometries. The parallelogram neutral n.2.2.A (C2h, 1Ag) is 0.11 eV lower in energy than the planar quadrilateral isomer n.2.2.B (Cs, 1A’). In the cationic state, our G4 calculations result in a near degeneracy of both cations c.2.2.A (Cs, 2A’) and c.2.2.B (C2v, 2A1). The isomer c.2.2.A in which one Mg-atom connects with two Si-atoms whereas the other Mg bonds with only one Si-atom is the most stable isomer but with an energy gap of only 0.07 eV below c.2.2.B. Si3Mg2. For this pentaatomic size, the neutral isomer n.3.2.A (Cs, 1A’) is formed by adding two Mg-atom on two edges of the Si3 triangle. The second isomer n.3.2.B, which is generated by replacing two positions of Si atoms in Si5 framework39 by two Mg-dopants, lies at 0.13 eV above n.3.2.A. In the cationic state Si3Mg2+, the planar structure c.3.2.A (Cs, 2A’), being the corresponding cation of n.3.2.A, is 0.22 eV lower in energy than the next isomer c.3.2.B.



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Si4Mg2. The neutral Si4Mg2 n.4.2.A (C2v, 1A1) can be considered as a substitution of two Siatoms of the Si6 counterpart39 by two Mg-atoms. In other words, it can be formed by replacing one Si-atom of the trigonal bipyramid Si5 by one Mg-atom whereas the remaining Mg-atom caps on a face of it. The higher symmetry n.4.2.B (D2d, 1A1), being 0.30 eV higher in energy than n.4.2.A, becomes the second stable isomer. In the cationic state, c.4.2.A (C2, 2A) is the corresponding cation of n.4.2.A and 0.38 eV more stable than the next isomer c.4.2.B. Si5Mg2. The neutral isomer n.5.2.A (C2, 1A) possesses the form of the pure Si6 skeleton in which one Mg atom changes one Si position, whereas the other Mg caps on one of its faces. Two next isomers n.5.2.B and n.5.2.C, being 0.18 eV higher in energy than n.5.2.A, exist as degenerate equilibrium minima. In the shape of n.5.2.B, both Mg atoms substitute two positions of Si atoms in pentagonal bipyramid Si7 while n.5.2.C is formed by adding both Mg atoms on the trigonal bipyramid Si5. In the cationic state, as in Si3Mg2 and Si4Mg2, the corresponding cation of the most stable neutral Si5Mg2, c.5.2.A (C1, 2A), which lies at 0.26 eV lower in energy than c.5.2.B, remains the most stable cationic structure. Si6Mg2. Our G4 results emphasize two degenerate structures n.6.2.A (Cs, 1A') and n.6.2.B (C2, 1

A) for the neutral Si6Mg with an energy separation of only 0.04 eV. Both of them are generated

by capping two Mg atoms on two different positions of the Si6 framework. However, it can also be considered that n.6.2.A is formed by replacing a Si atom in pentagonal bipyramid Si7 skeleton by one Mg atom whereas the other Mg adds on a face of it. The isomer n.6.2.C is also quite stable, being at 0.14 eV above n.6.2.A. In the cationic state, again c.6.2.A (Cs, 1A') is the corresponding cation of the lowest-lying neutral n.6.2.A. However, c.6.2.C, the corresponding cation of the lowest-lying neutral n.6.2.B, is much less stable than c.6.2.A with a gap of 0.31 eV. The second isomer c.6.2.B (C2h, 2Bu),


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being 0.14 eV higher in energy than c.6.2.A, is generated by substituting two Si atoms in bicapped octahedral shape of neutral Si8 39 by two Mg atoms. Si7Mg2. Structure n.7.2.A (C2v, 1A1) can be considered as arising from replacement of two Siatoms of the pentagonal bipyramid Si7 framework by both Mg-atoms, while two Si atoms adsorb on the pentagonal faces. The next isomer n.7.2.B, being 0.33 eV higher in energy, possesses the bicapped octahedral shape of the pure neutral Si8 in which one Si position is substituted by an Mg-atom whereas the remaining Mg-atom makes a bridge between two other Si atoms. The isomer n.7.2.C, which is only 0.01 eV above n.7.2.B, keeps the pentagonal bipyramid Si7 shape, in which both Mg atoms cap on it. In other words, n.7.2.C is characterized by the edge-capped pentagonal bipyramid of the cation Si8+ 41 where the first Mg atom takes the capping position and the remaining Mg adds on its face. However, the cation c.7.2.A, which is the corresponding cation of n.7.2.C, becomes the most stable structure being 0.17 eV below the second cationic isomer. Si8Mg2. The high symmetry n.8.2.A (D2d, 1A1) has both Mg atoms added on two opposite faces of the anion Si8- framework.39 The next isomer n.8.2.B, being 0.21 eV higher in energy, also has one Si8- counterpart in which one Mg caps on a face, and the other Mg adds on an edge. On the other hand, both n.8.2.A and n.8.2.B can be formed by replacing a Si atom in bicapped pentagonal bipyramid Si9 39 by an Mg atom whereas the remaining Mg adds on a face or an edge. The remaining isomers are much less stable. The cation c.8.2.A, being the corresponding cation of n.8.2.A, is also the lowest-lying isomer. However, our G4 calculations point out that it is degenerate with c.8.2.B with an energy gap of only 0.06 eV. The latter isomer can be generated by a substitution of both Mg atoms by two Si atoms in the pentagonal bipyramid Si7 in which three other Si atoms cap on different positions. The isomer c.8.2.C, being the corresponding cation of n.8.2.B, is lying 0.12 eV above c.8.2.A.



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Si9Mg2. Four lower-lying isomers of the neutral Si9Mg2 have the tetra-capped trigonal prism framework of Si10 in which an Mg atom replaces one of Si atoms whereas the other Mg caps at different positions. n.9.2.A (Cs, 1A') is found at 0.15 eV more stable than the second isomer possessing higher symmetry n.9.2.B (D3h, 1A1'). However, in the positively charged states, c.9.2.A, which is actually the corresponding cation of n.9.2.B, turns out to be the most stable isomer. Again, our G4 results indicate a near degeneracy of both cations c.9.2.A and c.9.2.B with an energy separation of only 0.08 eV. Si10Mg2. According to G4 calculations, the isomer n.10.2.A, which is formed by assembling two trigonal bipyramids Si5 and the Mg atoms are operated as linkers, each Mg linker connecting a top of bipyramid with an edge of another bipyramid, having thus a cyclic Si5-Mg-Si5-Mg shape, turns out to be the lowest-lying neutral structure of Si10Mg2. All the four next isomers n10.2.B, n10.2.C, n10.2.D and n10.2.E are higher in energy than n.10.2.A within a range from 0.16 eV to 0.24 eV. The latter four isomers can be formed by replacement of two Si atoms of the same hexa-capped trigonal prism Si12 cage43 by two Mg dopants, even though their geometric structures are significantly distorted. In the cationic state, the high symmetry isomer c.10.2.A (C2h, 2Bu) is located at 0.10 eV lower in energy than c.10.2.B. The c.10.2.A also exhibits the hexa-capped trigonal prism Si12 framework in which both Mg dopants actually substitute two Si atoms, whereas the c.10.2.B is formed by adding both Mg atoms on the faces of the tetra-capped trigonal prism Si10 framework. The cation of the cyclic neutral n.10.2.A is found to be much higher in energy. 3.2 Equilibrium Growth Pattern of the SinMgm Clusters On the basis of the structural features of the most stable isomers identified above for all sizes considered, the growth pattern of the clusters SinMgm with n = 1 – 10 and m = 1 – 2 can be established following some trends:


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i) The growth sequence of the singly doped neutral SinMg is quite similar to the singly lithium doped neutral SinLi clusters.9 A SinMg structure is characterized by the Mg dopant which favors its addition on either an edge or a face of the anionic ground state structure Si n- framework. However, in both energetically degenerate lowest-lying isomers of Si8Mg, the Mg dopant substitutes a Si atom in the anion Si8- counterpart, whereas the replaced Si atom adds on its face. In other words, both structures can be regarded as to be formed by replacement of a Si atom in the pure Si9 framework by an Mg atom. ii) For the monocations SinMg+, there is a significant difference between the behaviors of both dopants Li and Mg. Accordingly, the Mg atom seems to cap on one of the edge or the face of the cationic Sin+ instead of the neutral bare Sin such as in the case of the Li dopant. No exception on this observed trend is found in the size range of n = 1-10. iii) The doubly magnesium-doped neutral SinMg2 clusters grow up following a similar pattern as in the other series of neutral SinX2 clusters including X = Li and Al as reported in our previous studies.9,13 Their general growth pattern is that one Mg atom substitutes into a position of the Sin+1, whereas the other Mg atom is usually added on an edge, or a face, of the existing cluster. However, the most stable structure of Si7Mg2 is formed by replacement of two Si-atoms of the pentagonal bipyramid Si7 framework by both Mg atoms while the two Si atoms adsorb to the pentagonal faces. The neutral Si10Mg2 presents a new structural motif, a cyclic Si5-Mg-Si5-Mg, in which Mg atoms serve as linkers of two Si5 blocks. The more classical isomer of Si10Mg2 in which both Mg dopants replace two Si atoms of the same hexa-capped trigonal prism Si12 cage, also constitutes another exception on the above observation. iv) Similarly, for the positively charged state, it can be considered that SinMg2+ cations contain the cationic Sin+1+ frameworks in which one Mg atom actually substitutes into one Si position, whereas the remaining Mg dopant caps on an edge, or a face of the Si cluster. Again, the cationic



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Si10Mg2+ cluster emerges as an exception, due to the fact that it also possesses the hexa-capped trigonal prism Si12 skeleton in which both Mg dopants substitute two Si atoms. 3.3 Thermochemical Properties Recent reports of the pure and doped silicon clusters indicated that the composite G4 approach is a suitable method to investigate their thermochemical parameters, due to the accuracy with respect to the limited experimental results.12,35 Therefore, we continue to use in this study the G4 protocols to predict the thermochemical properties of singly and doubly magnesium doped silicon clusters. Their total atomization energies (TAEs) and the enthalpies of formation at both 0 K and 298 K are summarized in Table 1. The calculated heats of formation at 0 K are used to evaluate the adiabatic ionization energies (IE), obtained from the energy difference between the neutrals SinMgm and their corresponding SinMgm+ cations. In general, the IE values are rather low, but there is no consistent trend in going from one dopant to two dopants. Table 2 presents the IE values of SinMgm in comparison with those of the pure silicon clusters in which the IE values of SinMg are smaller than the IE values of Sin and larger than the IE of SinMg2. In other words, the Mg dopants tend to decrease the IEs of the resulting Si clusters. 3.4 Relative Stability of Clusters In order to probe the inherent stability of the clusters considered, the average binding energies (Eb) of clusters are examined. The average binding energies (Eb) can conventionally be defined as follows (equations 1-4): Eb(SinMg) = [nE(Si) + E(Mg) – E(SinMg)]/(n+1)

(1)

Eb(SinMg+) = [(n-1)E(Si) + E(Si+) + E(Mg) – E(SinMg+)]/(n+1)

(2)

Eb(SinMg2) = [nE(Si) + 2xE(Mg) – E(SinMg2)]/(n+2)

(3)

Eb(SinMg2+) = [(n-1)E(Si) + E(Si+) + 2xE(Mg) – E(SinMg2+)]/(n+2)

(4)


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where E(Mg), E(Si), E(Si+) are the total energies of the Mg-atom, Si-atom and the cation Si+, respectively. For their part, E(SinMg), E(SinMg+), E(SinMg2), and E(SinMg2+) are total energies of the neutral SinMg, cationic SinMg+, neutral SinMg2 and anionic SinMg2+ structures, respectively. All the energetic parameters are obtained from G4 total energies. While the Eb values are given in Table 3, plots illustrating their evolution are depicted in Figure 5. The Eb values raise with increasing cluster sizes. For SinMg clusters, the Si10Mg in both neutral and cationic states consistently reveal the highest Eb values as compared to the smaller neutral and cationic singly doped species. For neutral SinMg2 clusters, the Si9Mg2 presents with the highest Eb value as compared to those of the remaining doubly doped species. In the series of cations considered, Si10Mg2+ get a maximum peak in the Eb plot that indicates its high thermodynamical stability. Although there are some deviations at certain sizes from a monotonically increasing profile, such changes are rather small, being within the error margins of computations, to attach any particular importance. In order to show a larger picture of the doping effects on thermodynamic stabilities, Figure 6 displays a comparison of the binding energies of the neutral Sin, SinMg, SinLi, SinAl and SinB using G4 energies in which the binding energies of SinMg turn out to be the smallest ones, and those of SinB are the largest. It appears that while B marginally increases the cluster stability with respect to fragmentations, Mg, Li, and Al tend to decrease it. 3.5 Silicon Nanowires with Magnesium Linkers Having investigated the geometrical and energetic properties of the series of small SinMgm clusters, we now carry out a combination of some Sin clusters, namely the Si3, Si5, Si7, Si8 and Si10, used as building blocks with the corresponding number of Mg atoms and cations playing the role of linkers. These pure silicon units are chosen due to their high stabilities and their unchanged geometrical motifs following doping by Mg atoms.



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Our calculated results point out that addition of successive Sin units in a symmetrical way leads to formation of magnesium-linked silicon nanowires. By this way, we use different pure silicon clusters Sik with k = 3, 5, 7, 8, 10, as starting blocks and Mg atoms as linkers to generate the magnesium-doped silicon nanowires (SikMg)l. Geometry optimizations lead to two different forms that are linear and cyclic structures. Geometry optimizations of (SikMg)l structures with all real vibrational frequencies at the B3LYP/6-311G(d) level indicate that they are effectively characterized as equilibrium structures. Due to our computational limitations, however, we are not able to carry out optimizations with l > 6 for (Si3Mg)l, (Si5Mg)l and l > 4 for (Si7Mg)l, (Si8Mg)l and (Si10Mg)l. The shapes of equilibrium structures of both linear and cyclic forms of (Si3Mg)l, (Si5Mg)l, (Si7Mg)l, (Si8Mg)l and (Si10Mg)l are shown in Figures 7, 8, 9, 10, and 11, respectively. Cartesian coordinates of the optimized structures are listed in Table S2 of the ESI file. Some characteristic properties of these structures emerge as follows: (Si3Mg)l. The linear (Si3Mg)l are formed by assembling Si3 units and Mg atoms operated as linkers, each Mg connecting an edge of a triatomic Si3 and a Si atom of adjacent trimer. In the cyclic forms of (Si3Mg)l, each Mg linker connects two tops of adjacent triangles (Figure 7). We note that the dimer (Si3Mg)2 is an isomer of the series Si6Mg2. In order to compare their stabilities cautiously, the relative energies of the nanowires (Si3Mg)2, including both linear and cyclic forms, and the Si6Mg2 clusters are determined. G4 energies indicate that both cyclic form and linear form of (Si3Mg)2 are ~1.8 and 3.8 eV, respectively, higher in energy than the most stable n.6.2.A of the size Si6Mg2. (Si5Mg)l. The linear nanowires (Si5Mg)l are formed by assembling the Si5 units along the main axis of the trigonal bipyramids and Mg atom operates as a linker connecting two Si atoms which


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are two tops of adjacent bipyramids whereas in the cyclic forms of (Si5Mg)l, each Mg linker connect a top of bipyramid with an edge of another bipyramid (Figure 8). Similar to (Si3Mg)l, the dimers (Si5Mg)2 have the same number of silicon and magnesium atoms of the Si10Mg2. As described in a previous section, G4 calculations indicate that the cyclic form n.10.2.A is ~0.16 eV lower in energy than the next cage structure n10.2.B of Si10Mg2, whereas the linear form is much less stable than n.10.2.A with energy gap of ~4.0 eV. It thus proves that the cyclic form of the doubly magnesium doped Si10 cluster (Si5Mg)2 can be a starting block for creating nanowire structures. (Si7Mg)l. Similar to (Si5Mg)l, the linear nanowires (Si7Mg)l are created by assembling the pentagonal bipyramid Si7 building blocks in which each Mg atom links two Si atoms located at two tops of the bipyramid. In the cyclic form, Mg atoms tend to favor a connection with a Si atom located at the pentagonal plane of a Si7 unit with one or two Si atoms which is a top or an edge of a pentagonal face of another one (Figure 9). (Si8Mg)l. Geometry optimizations reveal that the building block Si8 possesses the shape of Si8[ref. 39] for both linear and cyclic forms of (Si8Mg)l and the Mg cations act as a bridge linking two Si atoms of two different Si8 units or connecting one Si atom with an edge of another building block (Figure 10). (Si10Mg)l. According to our calculations, the (Si10Mg)l nanowires are formed by using the Si10 building blocks that are connected together by Mg linkers. These Si10 building blocks have the bicapped squared anti-prism cage of the pure Si102- dianion,44 except for the first Si10 unit in linear forms which is connected with only one Mg cation possesses the tetra-capped trigonal prism framework of Si10 (Figure 11).



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In an attempt to evaluate the relative stability of these assembled nanowire types, the average assembling energy of (SikMg)l forms are also investigated and summarized in Table 4. The average assembling energy is the energy released when the aggregate is formed from fragments per unit. This parameter provides us with an estimate about the tendency of assembling the cluster. The assembling energy of (SikMg)l formed from the Sik cluster and divalent linkers Mg are computed as follows (equation 5): EA[(SikMg)l] = {l E[SikMg] - E[(SikMg)l]} / l

(5)

Calculated results of the average assembling energy of (SikMg)l for the series of k = 5, 7, 8 and 10 show that silicon clusters Sik tend to be assembled in ring forms (Rl) over the linear forms (Ll) as the assembling energy of the Rl are larger than those of the Ll. However, the average assembling energy of the linear form appear to be increased with the size, meaning that linear nanowire becomes also a real possibility when the size l of the chain can significantly be larger than those considered here. 4. Concluding Remarks In the present theoretical study, we investigated the structures of the singly and doubly magnesium doped silicon clusters in both neutral and cationic states SinMgm with n = 1-10 and m = 1-2. The total atomization energies, heats of formation and binding energies were determined using the composite G4 protocol. The binding energies of the Mg-doped Si clusters are decreased with respect to the pure Si counterparts, irrespective of the charge state. As there is no experimental value available for these systems, the computed results can be regarded as predicted values, with an expected error margin of ± 3 kcal/mol (± 0.15 eV or ± 12 kJ/mol), due to the uncertainty of the experimental value of the heat of formation of the silicon atom and the inherent accuracy of the G4 protocol.


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The growth sequence of the singly doped neutral SinMg is similar to that of the singly doped neutral SinLi clusters. In SinMg structures, the Mg atom tends to favour addition on either an edge or a face of the anionic ground state structure Sin- framework. Only in Si8Mg, Mg substitutes a Si atom in the Si9 framework. For the cations SinMg+, there is a difference between the behavior of Mg and Li. The Mg atom seems to cap on one of edge or face of the cationic Si n+ instead of the neutral bare Sin like as in the case of Li. The doubly Mg-doped neutral SinMg2 clusters grow up basically following a way comparable as the doubly doped neutral SinX2 with X = Li, Al reported in previous studies. Their growth pattern is that one Mg atom substitutes into a position of Sin+1, whereas the other Mg atom is usually added on an edge, or a face, of the existing cluster. There are however a few exceptions of this observation such as Si10Mg2 whose lowest-lying isomer has a cyclic Si5-MgSi5-Mg motif. Similarly, SinMg2+ cations contain the cationic Sin+1+ frameworks in which one Mg atom actually substitutes into a Si position whereas the remaining Mg atom caps on an edge, or a face. Again, Si10Mg2+ appears as an exception of this trend. The most interesting result of this study is that the Mg dopant, due to its large electron transfer capacity, behaves as a cation Mgδ+ and thereby induces an ionic entity with the Sinδanionic partner. The resulting Mg cation can be served as a linker between Sikδ- blocks leading to stabilized linear and cyclic [(Sik)Mg]l structures. In fact, the higher stability of the cyclic dimer (Si5Mg)2 over more conventional isomers tends to point out the role of Mg as linkers. In the systems with k = 5, 7, 8 and 10, the linear frameworks can be regarded as promising starting blocks for silicon assemblies. Further studies on these binary clusters are highly desirable to probe their properties as potential 1D-nanowire materials.



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Supporting Information. Table S1 contains the G4 total energies of SinMgm0/+ clusters and Table S2 lists Cartesian coordinates of silicon nanowires with magnesium linkers (SikMg)l (k = 3, 5, 7, 8, 10, l = 2 – 6 for k = 3 and 5; l = 2 – 4 for k = 7, 8, and 10) optimized at B3LYP/6311G(d) level. This material is available free of charge via the Internet at http://pubs.acs.org. Acknowledgments: NMT thanks Ton Duc Thang University for support. MTN is indebted to the KU Leuven Research Council (GOA and PDM programs). We are also grateful to Dr. Vu Thi Ngan from Quy Nhon University for valuable assistance during the preparation of the manuscript.


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Table 1. Total Atomization Energy (TAE), Heats of Formation at 0K [∆Hf (0 K)] and 298K [∆Hf (298 K)] of SinMgm(n = 1–10, m = 1–2) in both Neutral and Cationic States Obtained Using G4 Calculations (kcal/mol)

Structure

TAE

ΔHf (0 K)

ΔHf (298 K)

n.1.1.A c.1.1.A n.2.1.A c.2.1.A n.3.1.A c.3.1.A n.4.1.A c.4.1.A n.5.1.A c.5.1.A n.6.1.A c.6.1.A n.7.1.A c.7.1.A n.8.1.A c.8.1.A n.9.1.A c.9.1.A n.10.1.A c.10.1.A n.1.2.A c.1.2.A n.2.2.A c.2.2.A n.3.2.A c.3.2.A n.4.2.A c.4.2.A n.5.2.A c.5.2.A n.6.2.A c.6.2.A n.7.2.A c.7.2.A n.8.2.A c.8.2.A n.9.2.A c.9.2.A n.10.2.A c.10.2.A

25.5 -132.5 121.8 -44.1 215.4 41.8 296.7 134.2 418.2 256.7 503.1 345.0 593.1 448.3 682.7 523.5 788.4 629.5 900.4 744.2 50.6 -84.0 137.3 -10.1 227.1 87.8 348.9 186.8 441.4 290.8 534.3 382.7 622.7 469.3 722.2 564.3 841.8 676.3 904.6 774.4

116.6 274.6 127.5 293.4 141.1 314.7 167.0 329.5 152.7 314.2 174.9 333.1 192.2 336.9 209.8 368.9 211.3 370.2 206.5 362.7 126.4 260.9 146.9 294.2 164.3 303.6 149.6 311.8 164.3 314.9 178.6 330.3 197.5 350.8 205.1 363.0 192.8 358.2 237.2 367.4

117.0 275.0 127.9 293.8 141.7 315.3 168.0 330.6 153.6 315.1 176.0 333.9 193.7 338.4 211.3 370.6 213.0 372.2 208.5 364.6 126.5 261.1 147.7 294.8 165.0 303.8 150.3 312.7 165.4 315.9 179.8 331.5 198.8 352.4 206.6 364.7 194.4 360.2 239.1 369.6



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Table 2. Adiabatic Ionization Energies (IE, eV) of Singly and Doubly Mg Doped Si Clusters in Comparison with pure silicon clusters Sin (G4 calculations) Sin (*)

The number of atoms, n

Sin-1Mg

Sin-2Mg2

2

6.85

-

7.89

3

7.19

5.83

8.29

4

7.52

6.39

8.00

5

7.05

6.04

8.17

6

7.00

7.03

7.76

7

6.86

6.53

8.02

8

6.28

6.58

7.11

9

6.90

6.65

7.72

10

6.89

6.85

7.95

11

6.77

7.18

6.70

5.65

7.39

12 (*) The values taken from Refs. 35


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Table 3. Average Binding Energies (Eb, eV) of SinMgm (n = 1 – 10, m = 1 – 2) using G4 Calculations n

SinMg

SinMg+

SinMg2

SinMg2+

1

0.55

1.20

0.73

1.50

2

1.76

2.08

1.49

1.93

3

2.33

2.49

1.97

2.39

4

2.57

2.79

2.52

2.71

5

3.02

3.21

2.73

2.96

6

3.12

3.30

2.90

3.09

7

3.21

3.45

3.00

3.17

8

3.29

3.43

3.13

3.26

9

3.42

3.54

3.32

3.41

10

3.55

3.67

3.27

3.48

Table 4. Average Assembling Energies (eV) for the Linear and Cyclic forms of (Si3Mg)l, (Si5Mg)l, (Si7Mg)l, (Si8Mg)l, and (Si10Mg)l calculated at the B3LYP/6-311G(d) Level (Si3Mg)l n Linear Ring

(Si5Mg)l

(Si7Mg)l

(Si8Mg)l

(Si10Mg)l

Linear

Ring

Linear

Ring

Linear

Ring

Linear

Ring

2 0.36

1.20

0.51

2.01

0.27

0.41

0.39

1.61

0.08

1.11

3 0.61

1.45

0.90

2.30

0.51

0.53

0.65

1.36

0.41

1.41

4 0.79

1.05

1.14

2.39

0.69

0.87

0.89

1.43

0.62

1.45

5 0.92

1.04

1.29

2.40

6 1.00

1.03

1.39

2.41



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Figure captions. Figure 1. Shapes, electronic states, and relative energies (ΔE, eV) of the lower-lying isomers SinMg with n = 1−7 at the (a) neutral and (b) cationic states. ΔE values are obtained using the composite G4 method. Figure 2. Shapes, electronic states, and relative energies (ΔE, eV) of the lower-lying isomers SinMg with n = 8−10 at the (a) neutral and (b) cationic states. ΔE values are obtained using the composite G4 method. Figure 3. Shapes, electronic states, and relative energies (ΔE, eV) of the lower-lying isomers SinMg2 with n = 1−7 at the (a) neutral and (b) cationic states. ΔE values are obtained using the composite G4 method. Figure 4. Shapes, electronic states, and relative energies (ΔE, eV) of the lower-lying isomers SinMg2 with n = 8−10 at the (a) neutral and (b) cationic states. ΔE values are obtained using the composite G4 method. Figure 5. Average binding energy (Eb, eV) of the SinMgm/0/+ (n = 1−10, m = 1−2) clusters using the composite G4 method. Figure 6. Average binding energy (Eb, eV) of SinX with n = 1-10, X = Li, Mg, B, Al clusters using the composite G4 method. (The G4 values of SinLi, SinAl, SinB obtained from Refs. 9, 12, 13, and 35). Figure 7. Linear (L1-L6) and cyclic (R1-R6) forms of [(Si3Mg)]l. Figure 8. Linear (L1-L6) and cyclic (R1-R6) forms of [(Si5Mg)]l. Figure 9. Linear (L1-L4) and cyclic (R1-R3) forms of [(Si7Mg)]l. Figure 10. Linear (L1-L4) and cyclic (R2-R4) forms of [(Si8Mg)]l. Figure 11. Linear (L1-L4) and cyclic (R2-R4) forms of [(Si10Mg)]l.


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n.1.1.A (3∑-, C∞v) 0.00

n.2.1.A (1A1, C2v) 0.00

n.2.1.B (3A1, C2v) 0.68

n.3.1.A (1A1, C2v) 0.00

n.3.1.B (3A’, Cs) 0.96

n.4.1.A (1A’, Cs) 0.00

n.4.1.B (1A’, Cs) 0.07

n.5.1.A (1A’, Cs) 0.00

n.5.1.B (1A1, C2v) 0.53

n.6.1.A (1A1, C2v) 0.00

n.6.1.B (1A’, Cs) 0.10

n.6.1.C (1A, C1) 0.48

n.7.1.A (1A1, C5v) 0.00

n.7.1.B (1A’, Cs ) 0.07

n.7.1.C (1A’, Cs ) 0.34

a/ The neutral SinMg clusters

c.1.1.A (2Π, C∞v) 0.00

c.2.1.A (2A1, C2v) 0.00

c.3.1.A (2A’, Cs) 0.00

c.3.1.B (2B2, C2v) 0.51

c.4.1.A (2A1, C2v) 0.00

c.4.1.B (2A’, Cs) 0.10

c.4.1.C (2A, C1) 0.12

c.4.1.D (2B2, C2v) 0.22

c.5.1.A (2B1, C2v) 0.00

c.5.1.B (2A1, C2v) 0.48

c.6.1.A (2B2, C2v) 0.00

c.6.1.B (2A’, Cs) 0.09

c.6.1.C (2A’, Cs) 0.10

c.7.1.A (2A1, C2v) 0.00

c.7.1.B (2A’, Cs) 0.57

b/ The cationic SinMg+ clusters Figure 1



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n.8.1.A (1A’, Cs) 0.00

n.8.1.B (1A, C1 ) 0.00

n.8.1.C (1A, C1 ) 0.22

n.8.1.D (1A1, C2v ) 0.30

n.8.1.E (1A1, C2v ) 0.30

n.9.1.A (1A, C1) 0.00

n.9.1.B (1A, C1) 0.05

n.9.1.C (1A’, Cs) 0.15

n.9.1.D (1A’, Cs) 0.17

n.9.1.E (1A’, Cs) 0.41

n.10.1.A (1A1, C3v) 0.00

n.10.1.B (1A’, Cs) 0.52

n.10.1.C (1A’, Cs) 0.68

n.10.1.D (1A, C1) 0.85

a/ The neutral SinMg clusters

c.8.1.A (2A’, Cs) 0.00

c.8.1.B (2A, C1) 0.18

c.8.1.C (2A, C2) 0.25

c.8.1.D (2A, C1) 0.25

c.8.1.E (2A”, Cs) 0.40

c.9.1.A (2A, C1) 0.00

c.9.1.B (2A’, Cs) 0.25

c.9.1.C (2A, C1) 0.27

c.9.1.D (2A’, Cs) 0.29

c.9.1.E (2A, C1) 0.34

c.10.1.A (2A’, Cs) 0.00

c.10.1.B (2A’, Cs) 0.26

c.10.1.C (2A’, Cs) 0.28

c.10.1.D (2A’, Cs) 0.33

c.10.1.E (2A, C1) 0.37

b/ The cationic SinMg+ clusters Figure 2 26 ACS Paragon Plus Environment

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n.1.2.A (1A1, C2v) 0.00

n.1.2.B (3B1, C2v ) 0.12

n.2.2.A (1Ag, C2h) 0.00

n.2.2.B (1A’, Cs) 0.11

n.3.2.A (1A’, Cs ) 0.00

n.3.2.B (1A1, C2v) 0.13

n.3.2.C (1A’, Cs) 0.14

n.4.2.A (1A1, C2v) 0.00

n.4.2.B (1A1, D2d) 0.30

n.5.2.A (1A, C2) 0.00

n.5.2.B (1A1, C2v) 0.18

n.5.2.C (1A, C1) 0.18

n.6.2.A (1A’, Cs) 0.00

n.6.2.B (1A, C2) 0.04

n.6.2.C (1A’, Cs) 0.14

n.7.2.A (1A1, C2v) 0.00

n.7.2.B (1A, C2) 0.33

n.7.2.C (1A’, Cs) 0.34

n.7.2.D (1A, C1) 0.37

n.7.2.E (1A1’, D5h) 0.86

a) The neutral SinMg2 clusters

c.1.2.A (2B1, C2v) 0.00

c.2.2.A (2A’, Cs) 0.00

c.2.2.B (2A1, C2v) 0.07

c.3.2.A (2A’, Cs) 0.00

c.3.2.B (2A1, C2v) 0.22

c.4.2.A (2A, C2) 0.00

c.4.2.B (2A’, Cs) 0.38

c.5.2.A (2A, C1) 0.00

c.5.2.B (2B, C2) 0.26

c.6.2.A (2A’, Cs) 0.00

c.6.2.B (2Bu, C2h) 0.14

c.6.2.C (2A1, C2v) 0.31

c.7.2.A (2A’, Cs) 0.00

c.7.2.B (2A’, Cs) 0.17

c.7.2.C (2A1’, D5h) 0.27

b) The cationic SinMg2+ clusters Figure 3 27 ACS Paragon Plus Environment


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n.8.2.A (1A1, D2d) 0.00

n.8.2.B (1A’, Cs ) 0.21

n.8.2.C (1A, C1) 0.52

n.8.2.D (1A, C2) 0.60

n.9.2.A (1A’, Cs) 0.00

n.9.2.B (1A1’, D3h) 0.15

n.9.2.C (1A’, Cs) 0.20

n.9.2.D (1A1, D2d) 0.30

n.9.2.E (1A’, Cs) 0.51

n.10.2.A (1A, C2h) 0.00

n.10.2.B (1A, C1) 0.16

n.10.2.C (1A’, Cs) 0.20

n.10.2.D (1A, Cs) 0.22

n.10.2.E (1A, C1) 0.24

a) The neutral SinMg2 clusters

c.8.2.A (2B1, D2) 0.00

c.8.2.B (2A’, Cs) 0.06

c.8.2.C (2A’, Cs) 0.12

c.8.2.D (2B, C2) 0.16

c.8.2.E (2A’, Cs) 0.35

c.9.2.A (2A”, Cs) 0.00

c.9.2.B (2A, C1) 0.08

c.9.2.C (2A’, Cs) 0.10

c.9.2.D (2A’, Cs) 0.16

c.9.2.E (2A, C1) 0.25

c.10.2.A (2Bu, C2h) 0.00

c.10.2.B (2A’, Cs) 0.10

c.10.2.C (2A’, Cs) 0.23

c.10.2.D (2A, C1) 0.26

c.10.2.E (2A1’, C3v) 0.28

b) The cationic SinMg2+ clusters Figure 4 28 ACS Paragon Plus Environment

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Figure 5



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Figure 6


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L2

L3

L4

L5

L6

R2 R3

R5

R4

R6

Figure 7



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L2

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L3

L4

L5

L6

R2

R3

R5

R4

R6

Figure 8 32 ACS Paragon Plus Environment

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L2

L3

L4

R2

R3

R4

Figure 9

L2

L3

L4

R2 R3

Figure 10


R4


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L2

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L3

L4

R2 R3

Figure 11


R4

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Theoretical Study of the SinMgm Clusters and Their Cations: Toward

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