Ti@Si16 - American Chemical Society

J. Phys. Chem. C 2011, 115, 335–350

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Study of the Structural and Electronic Properties of [Ti@Si16]n, [Sc@Si16K]n, and [V@Si16F]n (n e 9) Aggregates from First Principles M. B. Torres,† E. M. Fernańdez,‡ and L. C. Balba´s*,§ Departamento de Matema´ticas y Computacioń, UniVersidad de Burgos, 09006 Burgos, Spain, Instituto de Ciencia de Materiales de Madrid, CSIC, 28049 Madrid, Spain, and Departamento de Fı´sica Teo´rica, UniVersidad de Valladolid, 47011 Valladolid, Spain ReceiVed: July 18, 2010; ReVised Manuscript ReceiVed: NoVember 11, 2010

Recent experimental and theoretical work has established that the ground state and low-lying energy isomers of endohedral M@Si16 clusters (M ) Sc-, Ti, or V+) have a nearly spherical cagelike form, an atomic-like closed shell electronic structure, and a large highest occupied molecular orbital (HOMO)-lowest unoccupied molecular orbital (LUMO) gap of ∼2 eV, which suggests the use of these clusters as basic units (superatoms) in the assembly of optoelectronic materials. As a first step in that direction, in this work are studied, by means of first-principles calculations, the trends in the formation of [Ti@Si16]n, [Sc@Si16K]n, and [V@Si16F]n aggregates as their sizes increase (n e 9). We identify especially stable linear, planar, and three-dimensional patterns that can be used to grow low-dimensional periodical systems. When n g 2, the aggregates with greater binding energy result from the bonding of n supermolecular units having D4d symmetry for the M@Si16 cage, instead of the Frank-Kasper symmetry ground state of the basic superatom. Particularly interesting aggregates are (i) [Ti@Si16]n rings when n g 6, which can be grown as wires or nanotubes; (ii) rings and linear forms of [Sc@Si16K]n aggregates having a rich variety of nearly degenerate isomers differing in the bonding site of K atoms and strongly varying electric dipole moments; and (iii) [M@Si16X]3m wires (m ) 1-3) formed by vertically stacking the [M@Si16X]3 starlike trimer with rotation of 60° between consecutive trimer units, which show interesting magnetic configurations for M ) V and X ) F. The HOMO-LUMO gap for the most favorable structure decreases with size, and the aggregates become nearly metallic when n e 9. Introduction Interest in the study of aggregates built from small atomic clusters is motivated by their potential use as building blocks for new functional materials and devices at the nanoscale.1-4 To achieve this goal, it is important to investigate how the system geometry depends on the interparticle coupling and how it affects the physical properties of the systems. Chemically stable building blocks should have a closed electron configuration with a large energy gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). The other important factor determining the cluster stability is the atomic geometry. Thus, the cooperative effects between electronic and geometrical factors can provide a guiding principle for designing stable building-block clusters. In a previous work,5 we have determined from first-principle calculations the geometrical and electronic structure of several low-lying energy isomers of M@Sin clusters (M ) Sc-, Ti, or V+) over an n range of 14-18. We obtained good agreement with the experimental results reported by Nakajima and co-workers6-10 about the endohedral character and extra stability of M@Si16 clusters, as well as their electron affinity and HOMO-LUMO gap. For the ground state geometry of these 68 valence electron clusters, we obtained a distorted FrankKasper Td structure in agreement with previous calculations11-13 * To whom correspondence should be addressed. E-mail: balbas@ fta.uva.es. † Universidad de Burgos. ‡ CSIC. § Universidad de Valladolid.

Figure 1. Structure and symmetry of several low-lying energy isomers of M@Si16 clusters: FK* ) distorted Frank-Kasper; FK ) FrankKasper; Penta ) pentagonal; F-like ) fullerene-like. For each structure and impurity are given the total energy difference, in electronvolts, with respect to the lowest-energy configuration, as well as the HOMO-LUMO gap (electronvolts), and ordinal number of the isomer. The three first rows correspond to M ) Sc-, Ti, and V+, respectively, that is, M@Si16 clusters with 68 valence electrons. The fourth and fifth rows correspond to M ) Sc and V, respectively, that is, neutral clusters with 67 and 69 valence electrons, respectively. V@Si16+ and V@Si16 are not found within the C5V Penta configuration.

(see Figure 1). However, for the ground state of Sc@Si16, Ti@Si16-, and V@Si16 clusters, which have 67, 69, and 69 valence electrons, respectively, we obtained a geometry similar to that of the 16-IV isomer in Figure 1, with symmetry D4d, which was called fullerene-like (f-like) by Kumar and co-workers.11,12 This fact has important consequences for our study. It is interesting to observe that the calculated5 HOMO-LUMO gap of neutral Sc@Si16 and V@Si16, (0.36 and 0.65 eV, respectively, in their D4d ground

10.1021/jp1066742  2011 American Chemical Society Published on Web 12/23/2010

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J. Phys. Chem. C, Vol. 115, No. 2, 2011

state) is drastically reduced with respect to that of the anionic species Sc@Si16- and V@Si16- (1.04 eV for the D4d configuration in both cases). In a recent work, Kumar and co-workers14 reported a D4d symmetry for the empty cage Si16H16 cluster as well as for M@Si16H16 endohedral clusters for which M ) Cr, Mn, or Fe. Furthermore, we provided in ref 5 an interpretation of the electronic structure and orbital projected density of states (PDOS) of M@Si16 clusters in the context of the spherical shell model perturbed by the crystalline field of the underlying ionic geometry. That model rests on the following assumptions. (1) For an empty spherical cage, the favorable eigenstates have as few radial nodes as possible because their radial charge density must be concentrated around the spherical shell of atoms. Within the harmonic-oscillator notation, these eigenstates are classified as nl, where n denotes the number of radial nodes and l is the orbital angular momentum. (2) With regard to the l selection rule, only those O+(3) levels of the M endohedral impurity transforming in the same irreducible representation of the cluster point group can be mixed in a given bonding state. Consequently, the covalent bonding in M@Si16 clusters results from the hybridization of the Si empty-cage states and the valence states of the endohedral atom having equal angular momentum l, which is of d type for the HOMO of the lowlying energy isomers. This was illustrated in Figures 8 and 9 of our previous work5 for isomers 16-I and 16-II of Sc@Si16-, as well as isomers 16-I, 16-II, and 16-IV of V@Si16+. This picture has been confirmed by the recent angle-dependent photoelectron spectroscopy experiments of Lau and co-workers.15 Further confirmation of our model for the electronic structure of M@Si16 clusters could be provided by analyzing the photoionization cross section of inner hybrid states, in the manner described by Chakraborty and co-workers16 in a recent work about the structure of Xe@C60. The D4d isomer of V@Si16+ has a 0.50 eV smaller binding energy than the Td ground state and a smaller HOMO-LUMO gap (1.50 eV compared to 2.25 eV for the ground state), as well as larger average Si-V and Si-Si distances (2.91 and 4.11 Å, respectively, for the D4d isomer compared to 2.81 and 3.97 Å, respectively, for the ground state). The first unoccupied single-particle orbital of the D4d isomer has s-p character.5 An important difference between the D4d isomer and the others in Figure 1 is the fact that the orbitals with the largest excess of charge are those of p type instead of the d orbitals. Simultaneously, the s-type orbitals have a larger defect of charge than the other isomers, as deduced from the population analysis performed in our previous work5 (see Figure 7 of that reference). The construction of new optoelectronic materials via assembly of molecules of the type V@Si16X where X is a halogen atom or Sc@Si16Y where Y is an alkali atom was suggested by Nakajima and co-workers,8 assuming that the large HOMOLUMO gap of the ionic superatom can be maintained in the range of ∼2 eV when the supermolecule is formed and a solid phase can eventually be grown from them. On the other hand, Reis et al.17,18 have shown the possible survival under room conditions of a metastable crystal with hcp structure formed from the Td isomer (FK) of M@Si16 (M ) Ti, Zr, or Hf). Gueorguiev and co-workers19 have studied finite [M@Si12]n nanowires with a hexagonal cross section for n e 7 and M ) Fe, Ni, Co, Ti, V, and Cu, finding that the HOMO-LUMO gap decreases gradually toward metallic behavior. Pentagonal and hexagonal core-shell silicon nanowires with various core compositions, including 3d transition metal atoms, have been

Torres et al. investigated by Berkdemir and Gu¨lseren,20 who also revealed a metallic behavior in all the cases, but the most favorable cross section is hexagonal or pentagonal depending on the encapsulated atom. Leitsmann and co-workers21 have investigated the influence of substitutional and interstitial transition metal doping on the electronic and magnetic properties of silicon nanocrystals. They found a strong influence of electron correlation effects by comparing results from a semilocal GGA approach to exchange correlation with those from a GGA+U treatment. The GGA+U treatment enlarges the GGA fundamental gap because those fully occupied (empty) levels with strong 3d character of the impurity are shifted toward lower (higher) energy, while those 3d states mainly localized at the Si neighbors remain less influenced.22 Thus, the GGA+U treatment stabilizes high-spin configurations of Si nanostructures doped with impurities like Fe and Mn that have strongly localized 3d electrons.21-23 Nevertheless, for the impurities Sc, Ti, and V considered in this paper, which have fewer localized 3d electrons than Fe and Mn, these shifting effects should be less noticeable, as explained in a recent review by Zunger and co-workers.24 In this paper, the magnetic effects will be discussed only occasionally, basically for V-doped aggregates. As a step in exploring bulk phases of new materials composed of neutral M@Si16X entities, we study in this paper the trends in the equilibrium structures and electronic properties of [Ti@Si16]n, [Sc@Si16K]n, and [V@Si16F]n aggregates formed from those cagelike M@Si16 isomers shown in Figure 1. Computational Procedure We have used the density functional theory25 (DFT) code Siesta26 within the generalized gradient approximation as parametrized by Perdew, Burke, and Ernzerhof27 for exchangecorrelation effects. Spin effects, in a collinear formulation, have been taken into account, without fixing the multiplicity beforehand. Relativistic spin-orbit interactions have not been considered in this work, because their effect on the magnetic properties of silicon clusters doped with the early 3d elements Sc, Ti, and V is expected to be weak. Details about the pseudopotentials and basis sets are the same as in our previous work.5 Specifically, we used norm-conserving scalar relativistic pseudopotentials28 in their fully nonlocal form,29 generated from the atomic valence configuration 3s23p2 for Si (with core radii of 1.9 au for s and p orbitals) and the semicore valence configuration 4s23p63dn for Sc (n ) 1), Ti (n ) 2), and V (n ) 3) (all of them with core radii, in au, of 2.57, 1.08, and 1.37 for s, p, and d orbitals, respectively). For K (F), we used valence configuration 4s1 (2s22p5) with a core radius of 3.64 au (1.39 au) for all s, p, and d valence orbitals. The matrix elements of the self-consistent potential are evaluated by integration in a uniform grid with a double-ζ plus polarization (DZP) basis. These calculations employ a double-ζ basis s,p (for Si, K, and F) and s,p,d (for M), with single polarizations d (for Si, K, and F) and p (for M). These basis sets and pseudopotentials of M atoms were tested and used in previous works.26,30,31 Backlund and Estreicher have studied recently32 the interactions of a TM (Ti, Fe, and Ni) interstitial in Si with a preexisting vacancy by using the Siesta method with the DZP basis. These authors have compared the accuracy of Siesta to the plane-wave VASP approach for Fe-related defects in Si and found only very small differences.33,34 The grid fineness is controlled by the energy cutoff of the plane waves that can be represented in it without aliasing (120 Ry in this work). The equilibrium geometries result from unconstrained conjugate-gradient structural relaxation using the

[Ti@Si16]n, [Sc@Si16K]n, and [V@Si16F]n (n e 9) Aggregates DFT forces. We tested several initial structures per aggregate (typically more than 20) until the force on each atom was smaller than 0.010 eV/Å. Results and Discussion [Ti@Si16]n Aggregates. In this subsection is presented a selection of [Ti@Si16]n isomers resulting from the optimization of a large set of initial arrangements. First, we investigated systematically the n ) 2 configurations to select those that can serve as seeds in the construction of larger aggregates following well-defined patterns. As a second step, we studied n ) 3-9 aggregates constructed according to these patterns, particularly certain types of configurations that can be used to grow infinite wires and nanotubes. The binding energy of a [Ti@Si16]n aggregate is defined as the difference in the aggregate energy between it and that of n identical Ti@Si16 separated units: Eb(n) ) nE(Ti@Si16) - E([Ti@Si16]n). The magnetic moment of the [Ti@Si16]n structures considered in this paper is zero, except for a few cases. [Ti@Si16]2. A very important result is that those dimers formed by the bonding of two f-like Ti@Si16 units with D4d symmetry (16-IV isomer in Figure 1) have greater binding energy than those formed by two FK* (C3V) or two FK (Td) isomers. The different bonding between cages depending on the symmetry of the cage was already stressed in the work of Kumar and Kawazoe.11 These authors estimated a 1.35 eV binding energy for a (Zr@Si16)2 dimer formed by two D4d cages with a 2.42 Å Si-Si bond length, whereas it was only 0.05 eV for a (Ti@Si16)2 dimer formed by two FK cages with long intercage Si-Si bonds of 4.04 Å. In our optimizations, the binding energy of a (Ti@Si16)2 dimer composed of two D4d cages at 2.52 Å is 2.02 eV (see isomer 2-4 in Table 1), whereas for the (Ti@Si16)2 dimer formed by two FK* units bonded as in the work of Kumar and Kawazoe,11,12 we obtained a binding energy of 0.09 eV. Thus, the dimer formed from the f-like isomer of Ti@Si16 results in a greater amount being bound compared the amount of the dimer formed from the FK or FK* isomers. We will see in the subsections for M ) Sc and M ) V that the D4d isomer of M@Si16 is also the preferred one for the formation of [Sc@Si16K]n and [V@Si16F]n aggregates when n g 2. Notice in Figure 1 that the FK* (FK) isomer of Ti@Si16 is 0.40 eV (0.37 eV) more tightly bound than the D4d one, but for the neutral Sc@Si16 and V@Si16 superatoms, the D4d isomer is the ground state5 (see Figure 1). In the rest of the paper, we will consider only aggregates formed from the M@Si16 isomer with f-like D4d symmetry. It is worth mentioning here that the recent optimization of Si20M structures shows, when M ) Ti, an endohedral Ti in a D4d cage of 16 Si atoms bonded to a planar cluster of four Si atoms.35 Several [Ti@Si16]2 configurations are shown in Figure 2, and the corresponding binding energies, HOMO-LUMO gaps, and electric dipole moments are reported in Table 1, together with the types of bonding between the two D4d cages. The 16 Si atoms of a M@Si16 unit with D4d symmetry divide into two groups, each one containing eight equivalent atoms: (i) those Si atoms, denoted here as Si1, that are in the vertices of two square faces arranged as a square antiprism and (ii) the other Si atoms, denoted here as Si2, that are in the apex of the pentagons whose bases are the edges of the two square faces. The a-b dimer configurations with higher binding energies are those with a pentagon of unit a parallel to other pentagon of unit b, glued by five Siai-Sibj bonds. For that type of configuration, denoted here as pentagon to pentagon (PTP), there are

J. Phys. Chem. C, Vol. 115, No. 2, 2011 337 TABLE 1: Binding Energies (Eb) with Respect to n Ti@Si16 Separated D4d Units, HOMO-LUMO Gaps, and Electric Dipole Moments for Several Low-Lying Energy Isomers of [Ti@Si16]n Aggregates (n ) 2-6)a n-isomer

bonding geometry

2-1 2-2 2-3 2-4 2-5 2-6 3-1 3-2 3-3 3-4 3-5 4-1 4-2 4-3 4-4 4-5 4-6 4-7 4-8 5-1 5-2 5-3 5-4 5-5 5-6 5-7 5-8* 5-9 6-1 6-2 6-3 6-4 6-5 6-6 6-7 6-8 6-9 6-10*

PTP-2 arc 2 PTP-1 PTP-0 STS-s chain 2 STS-t STS-e-i PTP-2 arc 3 PTP-2/PTP-1 STS-s chain STS-e star 3 STS-t chain 2×PTP-2 ∼ ring PTP-2 arc 4 2⊥STS-s chain 2 2|STS-s STS-e star 3 + 1 STS-s chain STS-t chain STS-e ring 4 5×PTP-2 ∼ ring 5 (3+2)PTP-2 ∼ ring 5 PTP-2 arc 5 2×STS-e star 3 bent STS-s chain STS-e star 3 + 1 + 1 2×STS-e star 3 specular -1 STS-e star 3 + 2 STS-e ring 4 + 1 2×(STS-e star 3) 60° PTP-2 ring 6 3×(PTP-2 arc2) ∼ ring 6 3|(STS-s chain 2) 2×(STS-s star 3) PTP-2 arc 6 2×PTP-2 arc 3 ∼ ring 6 3⊥STS-s chain 2 PTP-2 arc 2 + 2×STS-s 2×STS-s chain 3

Eb HOMO-LUMO (eV) gap (eV) dipole (D) 2.43 2.40 2.37 2.02 1.85 1.63 5.11 4.90 4.11 4.03 3.76 8.37 8.07 7.54 7.37 6.27 6.09 5.63 5.19 11.17 10.95 10.84 9.17 8.97 8.80 8.71 8.51 7.38 15.40 15.10 14.43 13.88 13.82 13.82 13.72 13.20 12.53 12.24

0.59 0.51 0.41 0.57 0.56 0.85 0.25 0.32 0.46 0.23 0.20 0.15 0.21 0.36 0.18 0.17 0.36 0.18 0.09 0.13 0.22 0.19 0.13 0.29 0.11 0.28 0.03 0.11 0.19 0.30 0.12 0.07 0.16 0.14 0.08 0.24 0.25 0.22

1.09 0.40 1.15 0.02 1.01 0.05 1.21 1.26 1.13 0.27 2.63 0.00 0.22 0.01 1.76 1.22 0.65 3.12 0.00 2.20 0.59 1.30 2.76 0.85 4.93 3.70 3.82 0.96 1.26 0.82 0.55 2.95 1.40 2.71 0.17 0.32 2.45 6.71

a The labels in the second column (bonding geometry entry) are explained in the text. The asterisk for isomers 5-8 and 6-10 means that they have magnetic moment 2 µB.

Figure 2. Several low-lying energy isomers of [Ti@Si16]2 aggregates formed from Ti@Si16 units with D4d symmetry, labeled according to Table 1.

three different cases depending on the number of Si1a-Si1b bonds, namely, two, one, and zero. These cases are denoted in Table 1 as PTP-2, PTP-1, and PTP-0, respectively, and are represented in Figure 2 with the labels 2-1, 2-2, and 2-3, respectively. Note that the bonded square faces of the PTP-2 structure form a small arc composed of two D4d units. That is the origin of the designation PTP-2 arc 2 for isomer 2-1 in Table 1. In the following subsections, we will consider that structure as the main pattern for constructing larger n aggregates.

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Another important type of equilibrium a-b dimer configuration results from the joining of two parallel square faces of units a and b by means of four Sia1-Sib1 bonds. That structure is denoted here as square to square (STS) (see isomers 2-4 and 2-5 in Figure 2). Isomer 2-4 shows two specular D4d units with respect to a perpendicular plane to the Ti-Ti axis and is denoted as STS-s in Table 1. Isomer 2-5 in Figure 2 results from rotation of a unit of the STS-s dimer 45° around the dimer Ti-Ti axis and is denoted as STS-t in Table 1. Alternatively, the rotated unit can be seen as a translation of the first unit along the molecular axis. We remark that the PTP and STS [Ti@Si16]2 dimers are metastable equilibrium configurations of systems containing 32 Si atoms and two Ti atoms. Thus, starting with a STS-s dimer [equilibrium Ti-Ti distance (7.37 Å)], we decrease its Ti-Ti distance along the Ti-Ti axis until, eventually, a D4d unit overlaps with the other one. Surprisingly, the two units turn back spontaneously to the equilibrium STS-s configuration even for initial overlapping configurations with a 3.6 Å Ti-Ti distance. For a shorter Ti-Ti distance, the two D4d units coalesce into a Ti2@Si32 equilibrium structure having a 3.44 Å Ti-Ti distance and a binding energy 0.57 eV higher than that of the PTP-2 arc 2 aggregate. On the other hand, the binding energy of the PTP-2 arc 2 configuration (isomer 2-1) is 9.50 eV when it is calculated with respect to separated Si32 and Ti2 systems, and the energy of Si32 is optimized from the ground state configuration given by Yoo and co-workers.36 A third type of bonding configuration results from connection of a square edge of unit a with a square edge of unit b by means of two Sia1-Sib1 bonds. That structure is denoted here as square edge to square edge (STS-e) when the D4d units are specular images with respect to a perpendicular plane to the Ti-Ti axis. If one of these units is rotated 180° around the Ti-Ti axis, the configuration denoted here as STS-e-i results (see isomer 2-6 in Figure 2 and Table 1). It is symmetrical with respect to inversion through the central point of the square formed by the two Sia1-Sib1 bonds. Thus, PTP and STS types of bonding provide the basic dimer structures for the construction of larger aggregates. In this context, is interesting to note that the single-wall silicon nanotubes (SWSNTs) with a pentagonal section are more tightly bound than those with a hexagonal or square section, and that these SWSNTs are possible metals, as studied by Bai and coworkers.3 In contrast, H-terminated silicon compact nanowires possess band gaps wider than those of diamond silicon crystals, as described by Bai and co-workers.3 [Ti@Si16]3 and [Ti@Si16]4 Aggregates. Figure 3 shows several [Ti@Si16]3 and [Ti@Si16]4 aggregates, and their corresponding binding energies, HOMO-LUMO gaps, and electric dipole moments are given in Table 1. The highest-binding energy isomer of [Ti@Si16]3 has a PTP-2 type of bond between D4d units leading to an arc configuration (see 3-1 in Figure 3). A spin isomer of isomer 3-1 was found with a magnetic moment of 2 µB and a HOMO-LUMO gap of 0.28 eV. Isomer 3-2 in Table 1 (not shown in Figure 3) is a PTP-2 arc dimer joined to another D4d unit by a PTP-1 type of bonding. Other configurations with deformed D4d units were also found, as in the case of dimers, but only aggregates that preserve the symmetry of the individual units will be considered here. Isomer 3-3 is a linear STS-s chain with a 1/3 eV smaller binding energy per D4d unit than the PTP-2 arc 3 aggregate. However, the HOMO-LUMO gap is ∼2 times larger. We will see that only STS-s aggregates forming linear chains maintain a non-negligible HOMO-LUMO gap as n increases. Isomer

Torres et al.

Figure 3. Selected geometries of [Ti@Si16]n aggregates when n ) 3-4, formed from the D4d isomer of Ti@Si16. Two views of isomer 4-3 are given: lateral (left) and top (right).

3-4 is a two-dimensional symmetric star (denoted STS-e star 3) formed by edge-to-edge Si1-Si1 bonds between adjacent units. Each branch of that star 3 trimer can be grown as a chain via addition of STS-s bonded units to form planar aggregates with three branches of variable lengths. An interesting linear wire can be grown by stacking several STS-e star 3 units along the vertical axis. Other vertical stacking of STS-e star 3 units is possible via alternation of a star 3 unit with another 60° rotated star 3 unit. Isomer 3-5 in Table 1 (not shown in Figure 3) is formed by stacking three D4d units along the Ti-Ti axis, resulting in a high dipole moment. The lowest-energy isomer of n ) 4 aggregates results from two PTP-2 arc 2 dimers joined by STS-e bonding (see isomer 4-1 in Figure 3). Alternatively, it can be seen as the bonding of two STS-e dimers joined by two PTP-2 bond motifs. This 4-1 isomer has an inversion center, two perpendicular mirror σV planes (and two dihedral planes), and a σh plane containing the Ti atoms. It looks like a noncircular ring that can be grown in the same plane or along the perpendicular axis by means of Si1-Si1 bonds or Si2-Si2 bonds, respectively. Isomer 4-2 in Figure 3, with a PTP-2 arc 4 configuration, is the natural twodimensional (2D) extension of arc 3 isomer 3-1. It has a near degenerate spin isomer (30 meV higher binding energy and a magnetic moment of 2 µB) that is not quoted in Table 1. Isomer 4-3 is formed by two STS-s chain 2 dimers (2-4 in Figure 2) whose Ti-Ti axes are perpendicular to each other, that is, with the four Ti atoms forming a tetrahedron. Isomer 4-4 is composed of two STS-s chain 2 dimers whose Ti-Ti axes are parallel to each other, that is, with the four Ti atoms forming a quadrilateral. Notice that one of the two STS-s dimers in the isomer 4-4 structure is twisted ∼45° (and slightly deformed) to double the number of Si2-Si2 bonds. Isomer 4-5 is composed of a STS-e star 3 trimer with an additional D4d unit joined to a branch by means of an STS-s type of bond. Isomers 4-6 and 4-7 are linear STS-s and STS-t chains, respectively (STS-t chain not shown in Figure 3). The metallic-like nonmagnetic 4-8 isomer is a symmetric cross composed of four STS-e linked D4d units and is denoted as STS-e ring 4 (see Figure 3). The largest HOMO-LUMO gap is obtained for the threedimensional (3D) (4-3 isomer) and one-dimensional (1D) (4-6 isomer) aggregates composed of two STS-s chain 2 dimers. The value (0.36 eV) is similar to that of the single STS-s chain 2

[Ti@Si16]n, [Sc@Si16K]n, and [V@Si16F]n (n e 9) Aggregates

Figure 4. Selected geometries of [Ti@Si16]n aggregates (n ) 5-6) formed from f-like (D4d) units.

dimer (0.37 eV). Instead, for the 2D configuration composed of two STS-s chain 2 dimers (4-4 isomer), the HOMO-LUMO gap is only 0.18 eV. On the other hand, the largest dipole moment corresponds to the linear chain formed by isomer 4-7. [Ti@Si16]5 and [Ti@Si16]6 Aggregates. Several equilibrium configurations of [Ti@Si16]5 aggregates are represented in Figure 4, and their electronic properties are listed in Table 1. The lowest-energy isomer (5-1) forms an irregular ring composed of five PTP-2 bonded D4d consecutive cages, except the first and last units, which are bonded to each other by an STS-e type of bond. This planar structure is related to nearly degenerate isomers 5-2 and 5-3. The structure of isomer 5-2 can be seen as a PTP-2 arc 3 (3-1 isomer) joined to a PTP-2 arc 2 (2-1 isomer) by STS-e bonding, and isomer 5-3 forms a PTP-2 arc 5 structure. Isomer 5-4 (2×STS-e star 3) is a 3D aggregate formed by two STS-e star 3 aggregates that share a central D4d unit. Isomer 5-5 is an STS-s linear chain. The planar structures of isomers 5-6-5-9 (not shown in Figure 4) are formed by STS bonding of D4d units to different branches of the star 3 aggregate (isomer 3-4) or to the STS-e ring 4 aggregate (isomer 4-8). A representative selection of n ) 6 aggregates is shown in Figure 4, and their electronic properties are given in Table 1.

J. Phys. Chem. C, Vol. 115, No. 2, 2011 339 The highest-binding energy isomer in Figure 4 is isomer 6-1 (labeled 2×STS-e star 3 60°). It is a 3D structure formed by superposition of two STS-e star 3 aggregates along the vertical axis with a rotation of 60° of one trimer with respect to the other. The same procedure without the 60° rotation leads to aggregate 6-5, which has a binding energy smaller than that of isomer 6-1 by 1.58 eV. Isomers 6-2 and 6-3 have a planar ring structure, and their binding energies differ by 0.67 eV. Whereas isomer 6-2 is a regular ring 6, isomer 6-3 is formed by three PTP-2 arc 2 dimers bonded by STS-e bonds plus an additional Si2i -Si2j bond between consecutive dimers. The average Si-Si bond distance in isomer 6-3 is 2.61 Å, to be compared with 2.41 Å for isomer 6-2. Two nearly degenerate spin isomers of nonmagnetic aggregate 6-3 have been found with magnetic moments of 4 and 2 µB, respectively, and larger dipole moments (1.24 and 0.57 D, respectively). One can imagine a class of nanotubes with a rich variety of magnetic structures formed by the simple stacking along the vertical axis of ring 6 spin isomers 6-2 and 6-3. Isomer 6-4 is composed of three STS-s chain 2 dimers laterally stacked, which are joined by two Si2-Si2 bonds between two D4d units. To achieve that double bonding, the central dimer is rotated 45° around the Ti-Ti axis. The structure of isomer 6-5 already described is nearly degenerate with a PTP-2 arc 6 planar configuration labeled as 6-6 in Figure 4, with an irregular ring structure (6-7 isomer) formed by two PTP-2 arc 3 aggregates bonded by Si1-Si1 bonds (STS-e bonding). Isomer 6-8, denoted as 3STS-s chain 2, is formed by three STS-s chain 2 dimers arranged perpendicularly with respect to one another, as the 4-3 isomer of Figure 4. Structures of both isomers 4-3 and 6-8 have spin isomers with magnetic moments of 2, 4, and 6 µB. Thus, this type of configuration can be systematically grown leading to linear wires with interesting magnetic properties. Planar isomer 6-9 can be seen as the STS-e bonding of a PTP-2 arc 2 dimer with two STS-s chain 2 dimers. It was obtained by relaxing an initial configuration formed from an STS-e ring 4 with two D4d units added at adjacent branches. Magnetic isomer 6-10 (µ ) 2 µB) results from the bonding of two STS-s trimer chains by means of three Si2-Si2 bonds. Other n ) 6 planar isomers are derived from the STS-e star 3 or the STS-e ring 4 aggregates by the STS bonding of additional D4d units at different positions, and they have binding energies ranging from 11.55 eV for isomer 6-11 to 9.50 eV for isomer 6-17. From these patterns can be formed different infinite monolayers. An STS-s linear chain with a binding energy of 10.01 eV and a HOMO-LUMO gap 0.21 eV was also found. [Ti@Si16]7, [Ti@Si16]8, and [Ti@Si16]9 Aggregates. In Table 2 are given the binding energies, HOMO-LUMO gaps, and electric dipoles of several [Ti@Si16]7, [Ti@Si16]8, and [Ti@Si16]9 aggregates formed from Ti@Si16 superatoms with D4d symmetry, and in Figure 5 are represented the equilibrium structures of some of them. For [Ti@Si16]7 aggregates, only planar isomer 7-1, with a perfect PTP-2 ring structure, and linear isomer 7-2, with a high dipole moment, are reported in Table 2. Isomer 8-1 is formed by superposition of two STS-e ring 4 structures along the vertical axis with one ring 4 rotated 45° with respect to the other. The binding energy is ∼4 eV higher than that of isomer 8-4, which has two STS-e ring 4 structures superposed vertically without rotation. Isomer 8-2 is a completely regular planar ring; that is, each one of its eight sectors spans a π/4 angle. That ring is metallic with a negligible dipole moment, and from it can be constructed nanotubes by stacking several units along the vertical axis. Isomer 8-3 results by

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TABLE 2: Binding Energies (Eb), HOMO-LUMO Gaps, and Electric Dipole Moments for Ring, Chain, and Wire Types of [Ti@Si16]n Aggregates (n ) 7-9) Formed from Ti@Si16 Units with D4d Symmetrya n-isomer

cage bonding geometry

7-1 7-2 8-1 8-2 8-3 8-4 8-5 9-1 9-2

PTP-2 ring 7 STS-s chain 7 2×(STS-e ring 4) 45° PTP-2 ring 8 4⊥(STS-s chain 2) 2×STS-e ring 4 STS-s chain 8 3×(STS-e star 3) 60° PTP-2 ring 9

a

Eb HOMO-LUMO (eV) gap (eV) dipole (D) 18.72 12.79 22.58 21.48 18.61 18.51 15.09 26.53 25.11

0.16 0.10 0.01 0.01 0.23 0.04 0.12 0.09 0.11

0.12 9.48 0.10 0.02 0.21 0.02 9.58 2.15 1.81

Some of them are represented in Figure 5.

Figure 5. [Ti@Si16]n (n ) 7-9) aggregates formed from Ti@Si16 units with D4d symmetry. For isomers 8-1, 8-3, and 9-1, two views are given (front and upper).

vertical superposition of four STS-s chain 2 dimers whose Ti-Ti axes are arranged perpendicularly. That structure was also found with magnetic moments of 2, 4, and 6 µB and binding energies (0.13, 0.20, and 0.26 eV, respectively) smaller than that of nonmagnetic isomer 8-3. Isomer 9-1 is formed via addition along the vertical axis of isomer 6-1 another 60° rotated STS-e star 3 aggregate. For that structure, we have found nearly degenerate magnetic isomers with magnetic moments of 2, 4, and 6 µB. These type of [Ti@Si16]3m aggregates (m g 2), which can be grown as metallic wires, are very stable. The planar ring formed by isomer 9-2 (not shown) is the larger ring that can be constructed without disturbing the PTP-2 type of bonding and can be grown vertically to form metallic nanotubes. Discussion of Trends with Size. We compare now the trends in binding energy, HOMO-LUMO gap, and dipole moment for the linear STS-s chains, ring, and arc planar structures as the size n of the aggregate increases. For the STS-s chains, the binding energy per D4d unit increases from 1.01 to 1.88 eV over the n range of 2-8 and saturates to a value ∼2 eV. The HOMO-LUMO gap decreases slowly in that interval and tends

to ∼0.1 eV, but the dipole moment dramatically increases when n ) 7-8. This is due to the redistribution of electrons inside of the initial and final units of these chains (see the Mulliken analysis below). For the planar ring structures in the n range of 4-8, the binding energy per superatom increases from 2.09 to 2.69 eV and tends to a saturation value of ∼2.8 eV. When n ) 8, the HOMO-LUMO gap decreases, the ring becomes metallic, and the dipole moment rapidly decreases. The planar arc structure becomes a ring when n ) 7. We remark that the Kohn-Sham HOMO-LUMO gap (εHOMO - εLUMO)LDA/GGA obtained from the eigenvalues of local (LDA) or semilocal (GGA) functional is severely underestimated with respect to the real HOMO-LUMO gap values, which can be obtained (for finite systems) as the difference (IP - EA)exact between the ionization potential (IP) and the electron affinity (EA) calculated with the exact exchange-correlation functional.37 When the HOMO-LUMO gaps are estimated from (IP EA)LDA/GGA calculations for a set of first row atoms and molecules, the mean absolute error (MAE) with respect to the experimental (IP - EA)exp is reduced by 1 order of magnitude compared to the MAE resulting from (εHOMO - εLUMO)LDA/GGA values.37 A better estimation results from the accurate prediction of the fundamental gap of semiconductors achieved by a recent hybrid functional.38 From Figures 1-3 of that work are obtained correcting factors to (εHOMO - εLUMO)GGA in the ranges of 1.5-1.1, 1.2-1.1, and 1.3-1.1 for one-, two-, and threedimensional infinite semiconductor systems, respectively. A slightly larger factor can be expected for finite semiconductor systems. Independently of the correction factors discussed above, we observe that the HOMO-LUMO gap values of those aggregates growing according to a given pattern decrease when their size increases. That can be seen, for example, in the series of 2-4, 4-3, 6-8, and 8-3 (formed by the transversal stacking of STS-s isomer 2-4), as well as the series of isomers with arc n or with ring n motifs [and similarly for (Sc@Si16K)n and (V@Si16F)n aggregates considered later]. To understand the tendency of large aggregates toward metallic behavior, we performed a Mulliken population analysis of isomer 2-4 (see the following below), the result being that several Ti@Si16 units become bonded covalently by s-type delocalized electrons at the surface of the superatoms, whereas the Si-Ti bonding in a single Ti@Si16 unit is due to the spd hybridization of the sd orbitals of Ti with the sp orbitals of the Si atoms, which confine most of the electrons inside the cage [see also Figure 13c in the subsection about (V@Si16F)n aggregates]. Thus, the bonding between superatoms is analogous to the bonding of s-valence atoms, like sodium or gold, to form clusters. The Mulliken population analysis shows that the number of valence electrons of each superatom in isomer 2-4 is conserved, as in a typical covalent diatomic molecule. Nevertheless, the internal charge of each cage is arranged as follows. (i) The Si1 atoms in the square at the opposite side of the bond between cages increase its nominal valence in 0.005 electron (e) per atom, and the Si2 atoms that are bonded to these Si1 atoms increase in 0.040 e per atom. (ii) The Si1 atoms involved in the bond between cages decrease its valence 0.112 e per atom, and the Si2 atoms that are bonded to these Si1 atoms increase its valence in 0.146 e per atom. (iii) The Ti atom decreases its nominal valence in 0.316 e. A detailed analysis for each type of orbital reveals that the s and p orbitals of Ti lose 1.616 and 0.068 e, respectively; on the other hand, the d orbital of Ti gains 0.832 e, and its P polarization orbital gains 0.536 e. On the other hand, the s orbitals of the 16 silicon atoms lose 11.682 e, whereas


J. Phys. Chem. C, Vol. 115, No. 2, 2011 341

TABLE 3: Total Energy Excess with Respect to the Ground State (ET), Energies of Binding of the K Atom to the Sc@Si16 Superatom (Eb), HOMO-LUMO Gaps, and Electric Dipole Moments for Optimized Sc@Si16K Molecules with Several Sc@Si16 Symmetriesa ET (meV) Eb (eV) HOMO-LUMO gap (eV) dipole (D)

FK* (C3V), K on S

FK (Td), K on T

Penta (C5V), K on P

f-like (D4d), K on P

f-like (D4d), K on S

40 2.79 1.86 13.45

170 2.64 1.88 15.54

0 2.83 1.73 11.08

210 2.49 1.01 11.69

360 2.34 0.93 12.51

a The lowest-energy structure for each case has the K atom bonded on the hollow site of different faces: S for square, T for triangular, and P for pentagonal.

their p and polarization P orbitals gain 5.811 and 6.191 e, respectively. These rearrangements of the superatom nominal valence suggest the formation of spd hybrid orbitals for the bonding of Ti with the Si atoms in that particular symmetry. [Sc@Si16K]n Aggregates. Sc@Si16K Supermolecule. To find low-energy configurations of the Sc@Si16K supermolecule, we have optimized all the geometries having the K atom bonded to the inequivalent sites of those Sc@Si16 clusters with the symmetry of isomers 16-I-16-IV of Sc@Si16- anions in Figure 1. The total energies with respect to the lowest-energy configuration, the binding energy of the K atom, the HOMO-LUMO gap, and the dipole moment of the lowest-energy isomer for each superatom geometry are given in Table 3. The lowestenergy isomer of these Sc@Si16K aggregates has the K atom capping the pentagonal face of Sc@Si16 with C5V symmetry. Notice that the Sc@Si16- anion with C5V symmetry has 30 meV of excess energy with respect to the FK* ground state of Sc@Si16- (see Figure 1). However, for the neutral Sc@Si16, the sequence of low-lying energy isomers is just the inverse,5 that is, f-like, C5V, FK, and FK* with increasing energy. [Sc@Si16K]2 Aggregates. The binding energy of [Sc@Si16K]n aggregates is defined by the relationship Eb(n) ) nE(Sc@Si16K) - E([Sc@Si16K]n), where E(Sc@Si16K) is the energy of a Sc@Si16K molecule with K bonded on the pentagonal face of the Sc@Si16 unit having D4d cage symmetry. It is understood that the identity of the Sc@Si16K units is maintained somehow in the aggregate. This definition of Eb(n) is due to the fact that the n g 2 [Sc@Si16K]n aggregates have the highest binding energy when each of their n Sc@Si16K units shows D4d symmetry for the Sc@Si16 superatom. This is a remarkable fact because the more stable Sc@Si16K supermolecule is formed from the Sc@Si16 superatom with C5V symmetry, as mentioned above. Thus, the binding energy of [Sc@Si16K]n aggregates given in the following subsections should be reduced by 0.21 eV per molecule when calculated with respect to n separated Sc@Si16K units in the ground state, that is, with K on the pentagonal face of a Sc@Si16 superatom with C5V symmetry. In Figure 6 are shown several low-lying energy isomers of [Sc@Si16K]2 aggregates. The corresponding binding energy, HOMO-LUMO gap, and dipole moment are given in Table 4, together with the type of bonding between D4d cage superatoms (second column) and the bonding sites of the K atoms (third column). The PTP-2 type of bonding between cage superatoms is preferable to the STS-s type. The K atoms are bonded to the cages preferably on pentagonal (P) faces instead of square (S) faces. For each particular cage bonding structure, the K bonding sites are labeled according the conventions specified in the footnote of Table 4. Nearly degenerate isomers 2-1 and 2-2 in Figure 6 differ mainly in the bonding site of the K atoms. The I site is preferred, and this tendency is mantained for the n g 2 aggregates. Isomer 2-2 is more symmetrical and has a larger HOMO-LUMO gap than isomer 2-1. Degenerate isomers 2-3 and 2-4 (as well as

Figure 6. Several low-lying energy configurations of [Sc@Si16K]2 aggregates.

isomers 2-5 and 2-6) differ in the bonding sites of the K atoms, which induces a huge difference between the dipole moments. Isomer 2-7 has the largest dipole moment of those given in Table 4. STS-s-bonded linear isomers 2-8-2-11 have smaller binding energies and HOMO-LUMO gaps than the PTP-2 configurations, but they show a similar abrupt change in the dipole moment depending on the K bonding sites. [Sc@Si16K]3 and [Sc@Si16K]4 Aggregates. Figure 7 shows several low-lying energy isomers of [Sc@Si16K]3 aggregates, and their binding energies, average Si1-Si1 bond lengths, HOMO-LUMO gaps, and dipole moments are listed in Table 4. The PTP-2 arc 3 structure with K bonded on pentagonal faces (isomer 3-1) has the largest binding energy. The same structure but with K bonded on square faces (isomer 3-3) has a 1.3 eV smaller binding energy and a HOMO-LUMO gap reduced by a factor of 2. Isomers 3-2 and 3-4, with an STS-e star 3 structure, have larger HOMO-LUMO gaps than PTP-2 arc structures. These isomers exhibit very different dipole moments because of the different K bonding sites (pentagon vs square face, respectively). In principle, it is possible to construct linear wires by stacking several units of isomers 3-2 and/or 3-4 along the vertical axis, having a rich variety of dipole moments depending on the relative K bonding sites. Isomers 3-5 and 3-6, with STS-s and STS-t structure, respectively, have smaller binding energies and HOMO-LUMO gaps than the other structures. Figure 8 shows several low-lying energy isomers of [Sc@Si16K]4 aggregates. Their binding energies, HOMO-LUMO gaps, and dipole moments are listed in Table 4. The PTP-2 arc

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TABLE 4: Binding Energies (Eb), HOMO-LUMO Gaps, and Dipole Moments for Several [Sc@Si16K]n Isomers (n ) 2-4)a 2-isomer

cage bonding structure

K bonding sites

Eb (eV)

HOMO-LUMO gap (eV)

dipole (D)

2-1 2-2 2-3 2-4 2-5 2-6 2-7 2-8 2-9 2-10 2-11 3-1 3-2 3-3 3-4 3-5 3-6 4-1 4-2 4-3 4-4 4-5 4-6 4-7 4-8 4-9 4-10 4-11 4-12 4-13 4-14 4-15

PTP-2 PTP-2 PTP-2 PTP-2 PTP-2 PTP-2 PTP-2 STS-s STS-s STS-s STS-s PTP-2 arc 3 STS-e star 3 PTP-2 arc 3 STS-e star 3 STS-s chain STS-t chain PTP-2 arc 4 2×PTP-2 ∼ ring 2×PTP-2 ∼ ring 2×PTP-2 ∼ ring 2×PTP-2 ∼ ring 2×PTP-2 ∼ ring 2⊥(STS-s chain 2) PTP-2 arc 4 2×PTP-2 ring STS-s chain 2⊥(STS-s chain 2) STS-s chain STS-e ring 4 STS-e ring 4 STS-t chain

P(e1/I) P(e2/e2) P(e1/e3) P(e1/e1) P(e1/i3) P(e1/i2) S(e1/e1) P(e1/i1) P(e1/e3) P(e1/e1) S(e1/e1) Pe1/I/If P(e1/e1/e1) S(e1/e1/e1) S(e1/e1/e1) P(i1/i3a/e1) P(i1/i1/e1) Pe1/I/I/I P(e1/e3/e1/e3) P(e1/e1/e1/e1) P(e1/i3/e1/i3) P(e1/i2/e1/i2) P(e1/e3/e3/e1) P(e1/e1/e1/e1) S(e1/e1/e1/e1) S(e1/e1/e1/e1) P(i1/i3a/i1/i3a) S(e1/e1/e1/e1) P(e1/i3/i1a/e3) P(e1/e1/e1/e1) S(e1/e1/e1/e1) P(i1/i1/i1/e1)

3.70 3.69 3.63 3.63 3.56 3.55 3.20 2.78 2.73 2.72 2.08 7.89 6.85 6.59 5.97 5.91 4.90 11.80 11.50 11.46 11.45 11.44 11.36 10.88 10.44 10.15 9.69 9.54 8.76 8.17 7.14 7.10

0.40 0.50 0.43 0.41 0.44 0.44 0.40 0.23 0.30 0.29 0.39 0.48 0.73 0.21 0.67 0.20 0.29 0.31 0.30 0.30 0.26 0.26 0.32 0.25 0.21 0.21 0.44 0.34 0.13 0.41 0.41 0.27

20.09 19.70 7.42 16.43 15.25 20.89 33.48 18.05 0.05 20.40 0.12 25.36 16.70 29.04 1.56 13.88 28.40 24.38 0.78 29.59 0.58 28.56 0.03 27.43 32.84 0.01 0.11 0.01 0.10 31.00 0.76 38.19

a These aggregates can be seen, alternatively, as being composed of two bonded Sc@Si16 superatoms, each having D4d symmetry, with two K atoms bonded on different sites. The labels in the second column indicate the type of bonding between the Sc@Si16 units. The labels in the third column refer to the bonding sites of K atoms separated by slashes from left to right: P (S) for pentagonal (square) face, I for intercage, e (i) for exterior (interior) face, and numbers indicating a given face, numbered according to a clockwise sequence around the PTP or STS bonding area. Contrary to an exterior face, an interior face has at least one Si atom involved in the PTP or STS bonding. The intercage bonding sites are classified as in front (If), “behind (Ib), or just in the plane (I) containing the two neighbor Sc atoms.

Figure 7. Several low-lying energy [Sc@Si16K]3 aggregates.

4 structure with the K atoms bonded on pentagonal faces has the largest binding energy. A similar structure with the K atoms bonded on square faces leads to isomer 4-8 (not shown in Figure 8), which has an ∼1.3 eV smaller binding energy and a very high dipole moment. Isomers 4-2-4-6 show a 2×PTP-2 ∼ ring structure with K atoms bonded on pentagonal faces. They are nearly degenerate, but the different K bonding sites induce huge changes in their dipole moments. Isomer 4-9 has a similar 2×PTP-2 ∼ ring skeleton but with the K atoms bonded on

square faces and an ∼1.3 eV smaller binding energy. Its dipole moment is near zero. Isomers 4-7 and 4-11 are composed of two STS-s dimers having the Sc-Sc axis mutually perpendicular, and with the Sc atoms forming a tetrahedron. Their binding energies differ by ∼1.3 eV because of the different K bonded sites, being pentagonal (4-7) or square (4-11) faces. That fact leads to very different dipole moments: 27.43 D for isomer 4-7 and 0.01 D for isomer 4-11. Both isomers are good candidates for growing linear wires. Isomers 4-10 and 4-11 are linear with an STS-s configuration whose HOMO-LUMO gaps change drastically with a slight change in the K bonding sites. Isomers 4-13 and 4-14 exhibit an STS-e ring 4 configuration, with the four K atoms on pentagonal and square faces, respectively. As seen before, when K atoms are on the square faces, the binding energy decreases and the dipole moment dramatically goes to near zero. The largest dipole moment is achieved for isomer 4-15 with an STS-t chain configuration. [Sc@Si16K]n (n ) 5-6) Aggregates. Figure 9 shows several low-lying energy configurations of [Sc@Si16K]n (n ) 5-6) aggregates. Their binding energies, average Si1-Si1 bond lengths, HOMO-LUMO gaps, and dipole moments are listed in Table 5. When n ) 5, the arc 5 types of configurations with K bonded on pentagonal faces (isomers 5-1-5-4) are more tightly bound than those irregular ring structures formed by joining a PTP-2 arc 3 trimer and a PTP-2 arc 2 dimer (isomers 5-5-5-7). As for n ) 2-4 aggregates, we see the strong


Figure 8. Several [Sc@Si16K]4 aggregates. Two views of threedimensional isomers 4-7 and 4-11 are given: lateral (left) and top (right).

dependence of the dipole moment and HOMO-LUMO gap on the K bonding sites. The 5-9 arc 5 (not shown) and 5-10 ring 5 isomers have the K atoms bonded on the external square faces and have smaller binding energies than an STS-s linear chain (not shown) with K bonded on pentagonal faces (isomer 5-8). A ring 5 structure similar to that of isomer 5-10 was not found for [Ti@Si16]5. For the n ) 6 isomers, the ring structures of 6-1 and 6-2 with K bonded on the pentagonal faces have larger binding energies than the arc 6 structures of 6-4-6-7. Similarly, the 6-8 ring 6 isomer is more stable than the 6-9 arc 6 isomer, both structures having the K atoms on the square faces. Ring isomers 6-2 and 6-8 exhibit the largest and smallest dipole moments, respectively, of all the planar aggregates considered in this section. The binding energy difference between nearly metallic isomers 6-1 and 6-2 is 0.53 eV (∼5 meV per atom), but their dipole moments differ dramatically. Interesting nanotubes with variable dipole moment can be formed by growing the 6-1 (or 6-2) skeleton along the vertical axis. Similarly, interesting wires can be formed by growing vertically isomers 6-3, 6-11, and 6-13. Three-dimensional isomer 6-3 has a binding energy between those of the ring 6 and arc 6 planar structures and shows the largest dipole moment of the [Sc@Si16K]n aggregates studied in this work, together with a non-negligible HOMO-LUMO gap of ∼0.5 eV. This isomer is formed from two STS star 3


Figure 9. Several [Sc@Si16K]n (n ) 5-6) aggregates. Two views of 3D isomers 6-3 and 6-11 are given: lateral (left) and top (right).

structures (3-2 trimer in Figure 7 with K atoms on pentagonal faces) stacked along the vertical axis with one unit rotated 60° with respect to the other. When the K atoms are bonded to square faces, isomer 6-13 is produced, with the binding energy and dipole moment drastically reduced, and a slightly larger HOMO-LUMO gap. Isomer 6-10 results from a stacking similar to that of isomer 6-3 but without the 60° rotation and with the K atoms bonded on the square faces, leading to a large reduction in the binding energy and dipole moment. The near metallic 6-11 isomer is a three-dimensional wire formed by the stacking of three STS chain 2 [Sc@Si16K]2 structures mutually perpendicular, as in [Sc@Si16K]4 isomer 4-11. Linear isomer 6-12 is reported here for the sake of comparison with 2D and 3D structures. [Sc@Si16K]7 and [Sc@Si16K]8 Aggregates. Figure 10 shows several low-lying configurations of [Sc@Si16K]n n ) 7-8 aggregates, and their binding energies, HOMO-LUMO gaps, and dipole moments are listed in Table 5. Like n ) 6 aggregates, the planar ring 7 configuration is more stable than the arc 7 one, and for each configuration, the dipole moment depends dramatically on the bonding sites of the K atoms. Linear STS-s chains when n ) 7-8 are less bound and become metallic. For [Sc@Si16K]8, it is no longer possible to obtain a stable arc 8 configuration. However, we optimized 3D isomer 8-4, composed of four STS-s chain 2 dimers stacked perpendicularly to one another, which can be grown as a longitudinal wire. It is a metallic wire with a negligible dipole moment.

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TABLE 5: Binding Energies (Eb), HOMO-LUMO Gaps, and Dipole Moments for Several [Sc@Si16K]n Aggregates (n ) 5-6) Shown in Figure 9a n-isomer


K bonding sites

Eb (eV)

HOMO-LUMO gap (eV)

dipole (D)

5-1 5-2 5-3 5-4 5-5 5-6 5-7 5-8 5-9 5-10 6-1 6-2 6-3 6-4 6-5 6-6 6-7 6-8 6-9 6-10 6-11 6-12 6-13 7-1 7-2 7-3 7-4 7-5 7-6 8-1 8-2 8-3 8-4 8-5

PTP-2 arc 5 PTP-2 arc 5 PTP-2 arc 5 PTP-2 arc 5 (3+2)(PTP-2) (3+2)(PTP-2) (3+2)(PTP-2) STS-s chain PTP-2 arc 5 PTP-2 ring 5 PTP-2 ring 6 PTP-2 ring 6 2×(STS-e star 3) 60° PTP-2 arc 6 PTP-2 arc 6 PTP-2 arc 6 PTP-2 arc 6 PTP-2 ring 6 PTP-2 arc 6 2×(STS-e star 3) 3⊥(STS-s chain 2) STS-s chain 2×(STS-e star 3) 60° PTP-2 ring 7 PTP-2 ring 7 PTP-2 ring 7 PTP-2 arc 7 PTP-2 arc 7 STS-s chain 7 PTP-2 ring 8 PTP-2 ring 8 PTP-2 ring 8 4⊥(STS-s chain 2) STS-s chain 8

Pe1/Ib/If/Ib/If Pe1/I/If/If/If Pe1/I/I/I/I Pe3/I/Pi2/If/I Pe1/I/Pi2/Pe3/I Pe1/I/Pi2/Pe1/I Pe3/Pi2/Pi3/Pe3/Pi2 P(e1/i3/i1a/i3/i1) 5(Se1) 5(Se1) Pi2/Ib/Pi2/Ib/Pi2/Ib 6(Pi2) 6(Pe1) Pe1/If/Ib/If/Ib/If Pe1/I/If/If/If/If Pe3/I/Ib/I/Ib/I Pe1/I/Pi2/Pi3/Pi2/I 6(Se1) 6(Se1) 6(Se1) S(e1/e1/e1/e1/e1/e1) P(e1/i3/i1a/i3/i1a/e3) 6(Se1) Pi2/I/I/Pi2/I/Pi2/I I/Pi2/Pi3/Pi2/Ib/Pi2/Pi3 7(Se1) Pe1/Ib/If/Ib/If/Ib/If Pe1/I/5If P(e1/i3/i1a/i3/i1a/i3/i1) 4(If/Ib) I/If/If/If/If/If/I/If 8(Se1) 8(Se1) P(e1/i3/i1a/i3/i1a/i3/i1a/i3)

15.64 15.64 15.49 15.43 15.14 15.11 15.01 12.75 12.74 11.06 20.80 20.27 19.81 19.50 19.50 19.48 19.11 18.14 16.82 15.53 15.45 14.84 9.68 26.78 26.57 23.97 23.35 23.32 17.98 31.28 31.18 27.61 23.15 21.02

0.30 0.30 0.14 0.16 0.21 0.24 0.26 0.33 0.06 0.13 0.09 0.03 0.46 0.22 0.13 0.14 0.13 0.10 0.05 0.31 0.15 0.13 0.57 0.19 0.17 0.14 0.19 0.19 0.02 0.29 0.25 0.24 0.07 0.03

17.58 28.86 20.18 17.15 13.05 31.17 33.57 16.65 26.32 0.15 3.32 41.41 44.49 10.40 28.71 18.39 11.03 0.05 12.61 5.19 0.09 0.74 0.53 19.38 4.71 0.15 8.18 31.03 21.23 0.65 31.12 0.03 0.02 2.66

a

The labels in the second and third columns are as described in the footnote of Table 4.

Figure 10. Several isomers of [Sc@Si16K]n (n ) 7-8) aggregates. Two views of isomer 8-4 are given.

[V@Si16F]n Aggregates. We have optimized all the possible V@Si16F configurations corresponding to an initial F atom upon the different sites of those V@Si16 isomers in Figure 1. The lowest-energy configuration shows the F atom bonded atop a Si1 atom of V@Si16 with D4d symmetry. That configuration is 0.25 eV (0.28 eV) more tightly bound than that with an F atom atop a Si atom of V@Si16 with FK* (FK) geometry. The HOMO-LUMO gap of the lowest-energy configuration of V@Si16F is 1.31 eV, that is, an intermediate value between those of the free V@Si16+ cation (1.50 eV) and neutral V@Si16 (0.65 eV), both of them with D4d symmetry. The binding energies, HOMO-LUMO gaps, electric dipole moments, and magnetic dipole moments of low-lying energy [V@Si16F]n (n ) 2-6 and 9) aggregates, all with D4d symmetry for the V@Si16 units, are given in Tables 6 (n ) 2) and 7 (n ) 3-9). The binding energy is calculated with respect to n separated V@Si16F molecules in the ground state, that is with F atop a Si1 atom of V@Si16 with D4d symmetry. Representative cases of these isomers are shown in Figures 11 (n ) 2), 12 (n ) 3-4), 14 (n ) 5), and 15 (n ) 6 or 9). For most of these aggregates, the bonding between D4d cages is of the PTP or STS-e type, leading to arc, star, and ring planar structures, as well as 3D arrangements that predominate when n g 5. The F atoms always bind on atop a Si1 atom in the square face not involved in the cage bonding, except for a few explicitly mentioned cases (isomers 2-1, 2-2, 2-3, and 5-2). The meaning of the labels for the F bonding sites in Tables 6 and 7 is explained in the footnote of Table 6. Many nearly degenerate isomers result from a change in a single F bonding



TABLE 6: Binding Energies (Eb), HOMO-LUMO Gaps, Electric Dipole Moments, and Magnetic Dipole Moments (µ) for Several Isomers of [V@Si16F]2 Aggregatesa n-isomer


F bonding site

Eb (eV)

HOMO-LUMO gap (eV)

dipole (D)

µ (µB)

2-1 2-2 2-3 2-4 2-5 2-6 2-7 2-8 2-9 2-10

PTP-0 PTP-0 PTP-0 STS-e-i STS-e PTP-1 STP PTP-2 PTP-0 PTP-0

i2/i2 o2/o2 i2/o2 i1/i2 i1/i2 i1/i2 o1/i2 i1/i2 Pi1/i2 Pi1/o2

1.82 1.63 1.63 1.57 1.54 1.52 1.50 1.44 1.43 1.40

0.55 0.51 0.52 1.14 1.14 0.90 0.65 0.75 0.53 0.59

4.07 2.77 2.64 0.60 0.18 0.40 3.18 0.91 2.17 1.51

0 0 0 0 0 0 0 0 0 0

a

Some of these aggregates are represented in Figure 11. The meaning of the labels for the F bonding sites in the third column is as follows. The two pairs of S1 atoms in the two diagonals of the square face define two types of F bonding sites, denoted as i or o if the diagonal is in plane or out of plane, respectively, with the neighbor V atoms. i1 (i2) refers to the first (second) Si1 atom starting clockwise from the D4d symmetry involved in the bonding; the o1 and o2 sites are defined similarly.

TABLE 7: Binding Energies (Eb), HOMO-LUMO Gaps, Electric Dipole Moments, and Magnetic Dipole Moments (µ) for Several Isomers of [V@Si16F]n (n ) 3-9) Aggregatesa n-isomer


F bonding sites

Eb (eV)

HOMO-LUMO gap (eV)

dipole (D)

µ (µB)

3-1 3-2 3-3 3-4 3-5 3-6 3-7 3-8 3-9 3-10 4-1 4-2 4-3 4-4 4-5 4-6 4-7 4-8 4-9 4-10 4-11 5-1 5-2 5-3 5-4 5-5 5-6 5-7 5-8 5-9 6-1 6-2 6-3 6-4 6-5 6-6 6-7 6-8 9-1

STS-e ∼ star 3 PTP-2 arc 3 PTP-2 arc 3 PTP-2 arc 3 STS-e star 3 STS-e star 3 PTP-2 arc 3 STS-e star 3 STS-e star 3 STS-e star 3 2×PTP-2 ∼ ring PTP-2 arc 4 2×PTP-2 ∼ ring 2×PTP-2 ∼ ring 2×PTP-2 ∼ ring 2⊥STS-s chain 2 PTP-2 arc 4 2⊥STS-s chain 2 2⊥STS-s chain 2 2×STS-s cross STS-e ring 4 2×PTP-2 ∼ ring + 1 (3D) 2×PTP-2 ∼ ring + 1 (2D) (3+2)(PTP-2) (2D) 2×PTP-2 ring + 1 (2D) PTP-2 arc 5 (2D) (3+2)(PTP-2) (2D) 2×PTP-2 ∼ ring + 1 (3D) (3+2)(PTP-2) (2D) PTP-2 ∼ ring 5 2⊥STS-e star 3 60° 2×PTP-2 ∼ ring + 1 + 1 PTP-2 ring 6 PTP-2 ring 6 PTP-2 arc 6 PTP-2 ring 6 PTP-2 arc 6 2×STS-e star 3 3⊥STS-e star 3 60°

i2/i2/i1 i1/i1/i1 i2/i1/i1 i1/i1/i1 i2/i2/i1 i2/i2/i1 i1/i1/i1 i1/o1/i1 i1/i1/i1 i1/i1/i1 o2/o1/o2/o1 4×i2 o1/o2/o1/o2 4×O1 4×O1 o1/o2/o1/o2 o2/o1/o2/o1 o1/o1/o1/o2 o1/o2/o2/o1 4×i1 4×i1 4×O1 + i1 4×O1 + o1_ 5×i2 4×O1 + o1 5×i2 i2/i1/i1/i1/i1 4×O1 + i1 i2/o2/i1/i1/i2 5×i2 3×i1 + 3×i1 6×i1 6×i1 o1/o2/o1/o2/o1/o2 6×i2 6×O1 o1/o2/o1/o2/o1/o2 2×(i2/i2/i1) 3×(i2/i2/i1)

4.26 4.05 4.00 3.91 3.80 3.72 3.70 3.66 3.60 3.54 6.50 6.28 6.27 6.26 5.95 5.77 5.74 5.63 5.47 4.84 4.72 9.23 9.05 8.86 8.78 8.59 8.43 8.35 8.32 7.87 13.11 12.92 11.85 10.87 10.78 10.67 9.80 9.69 22.70

0.46 0.48 0.22 0.25 0.36 0.35 0.21 0.21 0.28 0.24 0.53 0.24 0.35 0.19 0.16 0.25 0.21 0.14 0.33 0.38 0.30 0.19 0.35 0.27 0.25 0.22 0.22 0.20 0.19 0.29 0.16 0.03 0.21 0.30 0.11 0.32 0.13 0.12 0.03

2.67 4.34 3.63 2.25 2.35 2.33 2.57 1.93 0.51 0.52 1.84 2.58 0.52 4.98 4.17 1.10 3.14 1.87 0.83 0.27 0.91 8.33 8.73 0.42 8.33 5.76 1.68 7.41 1.89 0.62 3.73 4.68 0.17 0.12 5.06 6.76 1.41 2.64 2.80

0 4 2 2 2 0 2 2 2 0 0 2 4 4 2 4 2 2 0 2 0 4 0 6 2 4 6 4 6 0 6 4 4 0 6 0 4 8 12

a Some of them are represented in Figures 12 (n ) 3-4), 14 (n ) 5), and 15 (n ) 6 or 9). The labels in the second and third columns are as described in the footnote of Table 6.

site, for example, i1 to i2, or i1 to o1, etc. These changes dramatically affect the dipole and magnetic moments of the aggregate. [V@Si16F]2 Aggregates. Figure 11 shows several low-lying isomers of a [V@Si16F]2 dimer, and their binding energies, HOMO-LUMO gaps, and dipolar and magnetic moments are

listed in Table 6. The dimers with larger binding energies (1.63-1.82 eV) are those with PTP-0 bonding structure having four Si1-Si2 bonds and different F bonding sites. Those underlined sites in Table 6 that participate in the bonding between cage units belong to the square face. When the F atom is bonded to a Si2 atom, it is denoted with an initial P. This

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Torres et al.

Figure 11. Several isomers of [V@Si16F]2 aggregates.

happens for only PTP-0 isomers 2-9 and 2-10 (not shown) and is the main difference for isomers 2-1, 2-2, and 2-3. Isomers 2-6 and 2-8 exhibit PTP-1 and PTP-2 configurations, respectively. Isomer 2-7 shows a square to pentagon (STP) cage bonding structure. The PTP and STP structures, except the PTP-2 one, are not suitable for growing large aggregates (n g 3) with a well-defined trend. To achieve that goal, the PTP-2 (2-8) and STS (2-4 and 2-5) structures are more adequate. Isomer 2-4 shows a STS-e-i structure with the F atoms as far apart as possible, which impedes the formation of large linear chains. Similarly, nearly degenerate isomer 2-5 (binding energy 30 meV smaller than that of isomer 2-4) shows the two supermolecules as specular images of each other, and the F atoms are again far from each other. Both of these isomers have the higher HOMO-LUMO gap (1.14 eV) of the dimers. We have optimized several V2Si32F2 arbitrary compact configurations, composed of two V atoms, two F atoms, and 32 Si atoms, without resemblance to any kind of [V@Si16F]2 dimer with endohedral cage symmetry. The more tightly bound of these systems has a smaller binding energy (∼0.4 eV) with respect to two separated V@Si16F supermolecules than isomer 2-1. However, this fact does not exclude the existence of arbitrary V2Si32F2 configurations more stable than the dimers shown in Figure 11. [V@Si16F]3 Aggregates. Figure 12 shows several isomers of [V@Si16F]n (n ) 3-4) aggregates, and their binding energies, HOMO-LUMO gaps, and dipolar electric and magnetic moments are listed in Table 6. Apart from isomer 3-1, the PTP-2 arc 3 structures (isomers 3-2-3-4 and 3-7) are preferred to the STS-e star 3 ones (isomers 3-5, 3-6, and 3-8-3-10). Nonmagnetic isomer 3-1 shows an STS-e ∼ star 3 structure similar to that of isomer 3-5 but deformed by additional Si2-Si2 bonds. The 3-2, 3-3, and 3-7 arc 3 isomers show differences between their HOMO-LUMO gap values and dipole and magnetic moment values that are due to small structural differences. The magnetic 3-5 and nonmagnetic 3-6 star 3 isomers show a small difference in the V-V distances, which are slightly shorter for 3-6 than for 3-5. On the other hand, the difference between the dipole moment of 3-8 and those of 3-9 and 3-10 is due to the orientation of the F atoms in the 3-9 case, and to a small difference in size for the 3-10 case. From the STS-e star 3 isomers, it is possible to grow linear wires along their perpendicular axis. To understand the different magnetic moments and HOMO-LUMO gaps of structurally equivalent (nearly degenerate) isomers, in panels a and b of Figure 13 are compared the projected density of states (PDOS) on Si, V, and F atoms, as well as on 3d states of V atoms, for isomers 3-2 and 3-4,

Figure 12. Several isomers of [V@Si16F]n (n ) 3-4) aggregates.

respectively. We can see the strong hybridization of silicon sp orbitals with vanadium d orbitals, and that the cancelation of spin-up with spin-down V states is smaller for isomer 4-2 than for isomer 4-4. These facts are evident in the top part of panel c of Figure 13, where the number of up-down valence electrons is represented: notice the larger contribution of the central V atom to the magnetic moment, and the null magnetic moment of the V atom at the left side of the 3-4 aggregate. The small negative contribution of Si atoms to the magnetic moment is also noticeable. In the bottom part of panel c is shown the excess (defect) of total valence electrons for the Si (V) atoms in the left, center, and right units, with respect to its nominal valence. Upon addition of the excess of electrons on the F atoms, the excess or defect nominal valence electrons for the left, center, and right units of isomer 3-2 (3-4) are -0.030 (-0.030), 0.024 (-0.002), and 0.006 (0.032), respectively, which is related to the emergence of non-negligible dipole moments. In panel d of Figure 13 is represented the spatial distribution of the magnetization electron density of isomers 3-2 and 3-4 for an isosurface of 0.02 e/Å3, showing again the hybrid spd character of the bonds between V and Si atoms, and the s character of the Si-Si bonds between the units of (V@Si16F)3. Only small differences between the spatial magnetization of these isomers can be distinguished. [V@Si16F]4 Aggregates. In Figure 12 are shown several isomers of [V@Si16F]4. Their binding energies, HOMO-LUMO gapd, and dipolar electric and magnetic moments are listed in Table 7. Those configurations with the four V atoms in a plane, as isomers 4-1-4-5, are more stable than three-dimensional ones, as isomer 4-6 that has the four V atoms in the vertices of a tetrahedron. The ∼ring-type 4-1 and 4-3-4-5 isomers are formed by the STS-e bonding of two PTP-2 arc 2 dimers. Nonmagnetic isomer 4-1 has the largest binding energy and HOMO-LUMO gap. Small changes in the F bonding sites of



Figure 13. Several properties of isomers 3-2 and 3-4 of (V@Si16F)3 in Table 7. (a) Projected density of states of Si, V, and F (top) and of V atoms (bottom) in the left, center, and right units of isomer 3-2. (b) Same as panel a for isomer 3-4. (c) Number of up-down valence electrons (top) and total excess or defect of valence electrons (bottom) for the Si (red) and V (black) atoms in the left, center, and right units. (d) Up-down (magnetization) density of isomers 3-2 (top) and 3-4 (bottom) for isosurface densities of 0.02 e/Å3 (dark blue) and -0.02 e/Å3 (light red).

isomers 4-3 and 4-4 induce large differences in their HOMO-LUMO gap and dipole moment. On the other hand, the main structural difference among isomers 4-1, 4-3, 4-4, and 4-5 is the area of the rectangle formed by the V atoms, which is 59.14 Å2 for 4-1, 58.39 Å2 for 4-3, 56.28 Å2 for 4-4, and 55.48 Å2 for 4-5. Thus, the binding energy decreases when the area decreases. The differences between the arc 4 4-2 and 4-7 isomers are the bonding positions of the F atoms and the average Si-Si bonding distance, which is smaller for 4-2 than for 4-7. 3D isomers 4-8 and 4-9 are formed, as in the case of isomer 4-3 of [Ti@Si16]4, by two STS-s chain 2 dimers with the V-V axis perpendicularly arranged and the V atoms at the vertices of a (slightly deformed) tetrahedron. The V-V distances in the dimers of isomer 4-8 are 7.32 and 7.14 Å, to be compared with 7.15 and 7.14 Å for isomer 4-9. These structural differences

are responsible for the different magnetic moments, whereas the difference in dipole moments is due to the F bonding sites. Planar isomer 4-10 is formed by an STS-s dimer with two additional D4d units, each one bonded at opposite sides of the STS-s region by four Si1-Si1 bonds and two Si2-Si2 bonds. For planar isomer 4-11 (STS-e ring 4), the V atoms form a square and there are not Si2-Si2 bonds. Interesting rods can be constructed from that ring 4 structure by stacking several units along the vertical axis. [V@Si16F]5 Aggregates. Figure 14 shows several isomers of [V@Si16F]5, and their binding energies, HOMO-LUMO gaps, and dipolar and magnetic moments are listed in Table 7. The highest-binding energy isomer (5-1) exhibits a 3D structure that contains a V@Si16F molecule along the vertical axis of the planar 4-1 aggregate. Isomer 5-7 has a similar structure, with

348


Figure 14. Several isomers of [V@Si16F]5 aggregates. Two views of 3D isomer 5-1 are given: front and right.

small differences in the V-V distances of the rectangular pyramid skeleton. Planar isomer 5-2 contains a V@Si16F unit bonded along a coplanar axis of rectangular isomer 4-1. That nonmagnetic aggregate shows the largest HOMO-LUMO gap and dipole moment among those reported in Table 7. A similar structure is that of magnetic isomer 5-4. Isomer 5-3 is formed by an arc 3 structure joined to an arc 2 structure by the STS-e type of bonds and shows a high magnetic moment. Similar configurations are those of isomers 5-6 and 5-8, which differ from 5-3 mainly in the arrangement of F atoms. The more stable arc 5 configuration is that of isomer 5-5, with high dipole and magnetic moments. Isomer 5-9 can be seen as the joint of the first and last units of isomer 5-5, leading to a deformed ring 5 structure. [V@Si16F]6 and [V@Si16F]9 Aggregates. Figure 15 shows several isomers of [V@Si16F]6 and a representative [V@Si16F]9 isomer. Their binding energies, HOMO-LUMO gaps, and dipolar and magnetic moments are listed in Table 7. Isomer

Torres et al. 6-1 is formed by the stacking of two STS-e star 3 units (isomer 3-5 in Figure 12) along the vertical axis, with one unit rotated 60° with respect to the other. This structure shows a greatly enhanced binding energy and magnetic moment with respect to those of the single star 3 component (isomer 3-5). Isomer 6-2 is a 3D structure with the six V atoms forming a rectangular bipyramid and the F atoms pointing toward the center. It is metallic, with high magnetic and dipolar moments. The other n ) 6 isomers in Table 7 are planar with ring 6 or arc 6 structure. Changes in the bonding sites of the F atoms induce large changes in the dipole and magnetic moments of these isomers. Thus, a rich variety of nanostructures with interesting properties can be formed with these aggregates as basic units. Figure 15 also shows a 3D isomer of [V@Si16F]9 constructed from three star 3 trimers (isomer 3-5) stacked vertically with a rotation of 60° between each pair of trimer units. This metallic structure shows a binding energy and magnetic moment enhanced more than linearly with respect to those of isomers 6-1 and 3-5, composed of two star 3 components and one star 3 component, respectively. A magnetic isomer of 9-1 with a magnetic moment of 6 µB and a 0.35 eV smaller binding energy was also found. We see that, starting with the 3-5 trimer, it is possible to grow twisted metallic wires with high stability and very high magnetic moments. Summary and Conclusions M@Si16 structures (M ) Ti, Sc+, or V-) calculated previously5 show atomic-like features that are in agrement with experiments,8,15 as well as large HOMO-LUMO gap values (∼2 eV) that confer upon them potential interest as building blocks for optoelectronic materials with tailored properties. Although the ground state of Ti@Si16 has the Frank-Kasper Td structure, the (Ti@Si16)2 dimer is preferably formed with two f-like D4d units. That D4d cage is composed of two square faces and eight pentagons and contains only two types of nonequivalent Si atoms: those that are in the vertices of the square faces (Si1) and those that are not (Si2). The more stable V@Si16F

Figure 15. Several isomers of [V@Si16F]6 aggregates and a representative [V@Si16F]9 isomer. Two views of 3D isomers 6-1, 6-2, and 9-1 are given: front and right.



molecule is formed with the F atom bonded atop a Si1 atom of a square face of the V@Si16 isomer with D4d symmetry. On the other hand, the more stable Sc@Si16K molecule is formed with the K atom on the pentagonal face of the Sc@Si16 isomer with C5V symmetry. Nevertheless, the more stable [Sc@Si16K]2 dimer is formed with the Sc@Si16 units having D4d symmetry. The more favorable configurations for [Ti@Si16]2, [Sc@ Si16K]2, and [V@Si16F]2 aggregates result from two Si1-Si1 bonds and three Si2-Si2 bonds between the Si atoms of opposite pentagonal faces of two M@Si16 cages with D4d symmetry. The K and F atoms are not involved directly in the bonding. That dimer configuration, called here PTP-2 arc 2, is used to grow planar aggregates having arc geometry (n < 6) or ring geometry (n g 6). Nearly degenerate isomers of [Sc@Si16K]n ([V@Si16F]n) rings result from changes in the bonding site of K (F) atoms, but with very different electric and/or magnetic dipolar moments. By stacking these ring units along the vertical axis, we can grow nanotubes with electrical and magnetic properties varying over a broad range. Other types of [Ti@Si16]2, [Sc@Si16K]2, and [V@Si16F]2 structures, which are denoted here as the STS-s (STS-e) type of bonding, are formed by four (two) Si1-Si1 bonds between the Si atoms of square faces (edges) of two D4d units. From the STS-s dimer can be grown linear [Ti@Si16]n and [Sc@Si16K]n aggregates. More interesting are the one-dimensional [Ti@ Si16]2n, [Sc@Si16K]2n, and [V@Si16F]2n wires formed by stacking STS-s dimers with their M-M axis perpendicularly arranged, in such a form that the M atoms of two consecutive dimers form a tetrahedron. Several possibilities for bonding K or F atoms at that wire skeleton lead to nearly degenerate configurations with different electric and/or magnetic dipolar moments. The arc 3 trimer configuration arising from PTP-2 bonding is more favorable than the star 3 one provided by the STS-e type of bonding, with the M atoms forming an equilateral triangle. However, the stacking of these regular star 3 trimer units along the vertical axis, with a rotation of 60° between consecutive units, yields very stable [M@Si16X]3m aggregates, particularly those doped with Ti or V. The bonding of two PTP-2 arc 2 dimers by four STS-e Si1-Si1 bonds leads to a specular double arc 2 form (∼ring) that is the first isomer for [Ti@Si16]4 and [V@Si16F]4 and the second isomer for [Sc@Si16K]4 aggregates. For n ) 4 aggregates, these ∼ring and arc 4 configurations have comparable binding energies. The first isomer quoted for n ) 5 aggregates is an irregular PTP-2 ring 5 for [Ti@Si16]5, a PTP-2 arc 5 configuration for [Sc@Si16K]5, and a 3D configuration for [V@Si16F]5 formed by the [V@Si16F]4 ∼ ring 4 4-1 isomer plus an additional V@Si16F unit along the vertical axis. These structures are not adequate for the construction of large symmetric aggregates or wires. As mentioned above, the more favorable isomer of n ) 6 aggregates with Ti or V dopants is a 3D structure formed by the stacking of two star 3 structures with a rotation of 60° between both units. Both 3D and planar ring 6 isomers of [V@Si16F]6 show a rich variety of magnetic moments when the F atoms change their bonding sites and can be grown vertically. For [Ti@Si16]7, the arc 7 structure converges to the ring 7 one. Several planar ring 7 and arc 7 structures are reported for [Sc@Si16K]7. Nearly degenerate planar ring isomers of [Sc@Si16K]n (n ) 6-9) aggregates show dramatic change in their dipolar moments when the K atoms undergo slight changes in their bonding sites.

The vertical stacking of two star 4 structures rotated 45° is the first isomer of [Ti@Si16]8. That structure was not tested for Sc- and V-doped aggregates. [Ti@Si16]9 and [V@Si16F]9 structures formed by the vertical stacking of three star 3 isomers rotated 60° consecutively have a high binding energy, and for V-doped aggregates, they show high magnetic moments. These [V@Si16F]3m wire-type aggregates have magnetic moments of 2, 6, and 12 µB for m values of 1, 2, and 3, respectively. This sequence increases more than linearly with size, pointing to interesting molecular wire magnets. In conclusion, we have optimized several [Ti@Si16]n, [Sc@Si16K]n, and [V@Si16F]n structures for n e 9 with D4d symmetry for the M@Si16 unit. Particularly interesting aggregates are (i) planar rings for n g 6, which can be grown into nanotubes; (ii) 1D wires formed by stacking dimers with a mutually orthogonal M-M axis; (iii) wires formed by stacking the star 3 trimer, rotated 60° with respect to each other; and (iv) 1D wires formed by stacking star 4 n ) 4 aggregates along the vertical axis that are rotated 45° with respect to each other. In all these cases are found nearly degenerate isomers whose electric and magnetic moments depend dramatically on M dopant and K or F bonding sites. The HOMO-LUMO gap for the most favorable structure decreases with size, and the aggregates become quasi-metallic when n = 9. Trends in the size of aggregates with their geometrical structure as well as the binding energy, dipole moment, and other electronic properties have been discussed. Acknowledgment. We acknowledge the support of the Spanish Ministry of Science (Grant FIS2008-02490/FIS) and Junta de Castilla y Leoń (Grant GR120). References and Notes (1) Wu, Y.; Cui, Y.; Huynh, L.; Barrelet, C. L.; Bell, D. C.; Lieber, C. M. Nano Lett. 2004, 4, 433. (2) Wang, M. C. P.; Gates, B. D. Mater. Today 2009, 12, 34. (3) Bai, J.; Zeng, X. C.; Tanaka, H.; Zeng, J. Y. Proc. Natl. Acad. Sci. U.S.A. 2004, 101, 2664. (4) Claridge, S. A.; Castleman, A. W., Jr.; Khana, S. N.; Murray, C. B.; Sen, A.; Weiss, P. S. ACS Nano 2009, 3, 244. (5) Torres, M. B.; Fernańdez, E. M.; Balba´s, L. C. Phys. ReV. B. 2007, 75, 205425-1-12. (6) Ohara, M.; Koyasu, K.; Nakajima, N.; Kaya, K. Chem. Phys. Lett. 2003, 371, 490. (7) Koyasu, K.; Akutsu, M.; Mitsui, M.; Nakajima, N. J. Am. Chem. Soc. 2005, 127, 4998. (8) Koyasu, K.; Atobe, J.; Akutsu, M.; Mitsui, M.; Nakajima, N. J. Phys. Chem. A 2007, 111, 42. (9) Furuse, S.; Koyasu, K.; Atobe, J.; Nakajima, N. J. Chem. Phys. 2008, 129, 064311. (10) Koyasu, K.; Atobe, J.; Furuse, S.; Nakajima, N. J. Chem. Phys. 2008, 129, 214301. (11) Kumar, V.; Kawazoe, Y. Phys. ReV. Lett. 2001, 87, 045503. (12) Kumar, V.; Majunder, C.; Kawazoe, Y. Chem. Phys. Lett. 2002, 363, 319. (13) Reveles, J. U.; Khana, S. N. Phys. ReV. B 2006, 74, 035435. (14) Bahramy, M. S.; Kumar, V.; Kawazoe, Y. Phys. ReV. B 2009, 79, 235443. (15) Lau, T.; Hirsch, K.; Klar, Ph.; Langenberg, A.; Lofink, F.; Richter, R.; Rittmann, J.; Vogel, M.; Zamudio-Bayer, V.; Mo¨ller, V.; van Issendorff, B. Phys. ReV. A 2009, 79, 053201. (16) Chakraborty, H. S.; Madjet, M. E.; Renger, T.; Rost, J.-M.; Manson, S. T. Phys. ReV. A 2009, 79, 61201(R). (17) Reis, C. L.; Martins, J. L.; Pacheco, J. M. Phys. ReV. B 2007, 76, 233406. (18) Reis, C. L.; Pacheco, J. M. J. Phys.: Condens. Matter 2010, 22, 035501. (19) Gueorguiev, G. K.; Stafstro¨m, S.; Hultman, L. Chem. Phys. Lett. 2008, 458, 170. (20) Berkdemir, C.; Gu¨lseren, O. Phys. ReV. B 2009, 80, 115334. (21) Leitsmann, R.; Panse, C.; Ku¨wen, F.; Bechstedt, F. Phys. ReV. B 2009, 80, 104412.

350


(22) Leitsmann, R.; Ku¨wen, F.; Ro¨dl, C.; Panse, C.; Bechstedt, F. J. Chem. Theory Comput. 2010, 6, 353. (23) Ku¨wen, F.; Leitsmann, R.; Bechstedt, F. Phys. ReV. B 2009, 80, 045203. (24) Zunger, A.; Lany, S.; Raebinger, H. Physics 2010, 3, 53. (25) Kohn, W.; Sham, L. J. Phys. ReV. 1965, 145, 561. (26) Soler, J. M.; Artacho, E.; Gale, J. D.; Garca, A.; Junquera, J.; Ordejoń, P.; Sańchez-Portal, D. J. Phys.: Condens. Matter 2002, 14, 2745. (27) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. ReV. Lett. 1996, 77, 3865. (28) Troullier, N.; Martıńs, J. L. Phys. ReV. B 1991, 43, 1993. (29) Kleinman, L.; Bylander, D. M. Phys. ReV. Lett. 1982, 48, 1425. (30) Fernańdez, E. M.; Torres, M. B.; Balba´s, L. C. Int. J. Quantum Chem. 2004, 99, 39. (31) Torres, M. B.; Fernańdez, E. M.; Balba´s, L. C. Phys. ReV. B 2005, 71, 155412.

Torres et al. (32) Backlund, D. J.; Estreicher, S. K. Phys. ReV. B 2010, 81, 235213. (33) Sanati, M.; Gonzalez Szwacki, N.; Estreicher, S. K. Phys. ReV. B 2007, 76, 125204. (34) Estreicher, S. K.; Sanati, M.; Gonzalez Szwacki, N. Phys. ReV. B 2008, 77, 125214. (35) Willand, A.; Gramzow, M.; Alireza Ghasemi, S.; Genovese, L.; Deutsch, Th.; Reuter, K.; Goedecker, S. Phys. ReV. B 2010, 81, 201405R. (36) Yoo, S.; Shao, N.; Koehler, C.; Fraunhaum, Th.; Zeng, X. C. J. Chem. Phys. 2006, 124, 164311. (37) Cohen, A. J.; Mori-Sańchez, P.; Yang, W. Phys. ReV. B 2008, 77, 115123. (38) Brothers, E. N.; Izmaylov, A. F.; Normand, J. O.; Barone, V.; Scuseria, G. E. J. Chem. Phys. 2008, 129, 011102.

JP1066742

Ti@Si16 - American Chemical Society

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