- Email: [email protected]
Electron counting rules for transition metal-doped Si12 clusters
Accepted Manuscript Title: Electron Counting Rules for Transition Metal-Doped Si12 Clusters Author: Nguyen Duy Nguyen Tien Trung Ewald Janssens Vu Thi Ngan PII: DOI: Reference:
S0009-2614(15)00876-3 http://dx.doi.org/doi:10.1016/j.cplett.2015.11.025 CPLETT 33429
To appear in: Received date: Revised date: Accepted date:
21-7-2015 12-11-2015 16-11-2015
Please cite this article as: N. Duy, N.T. Trung, E. Janssens, V.T. Ngan, Electron Counting Rules for Transition Metal-Doped Si12 Clusters, Chem. Phys. Lett. (2015), http://dx.doi.org/10.1016/j.cplett.2015.11.025 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Highlights: - PSM and pELI-D allow counting of delocalized electrons for a doped Si cluster.
- The enhanced stability of Si12Cr is due to the 16-electron shell closure. - The enhanced stability of Si12Fe is due to the 18-electron shell closure.
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- 2Dz2 orbital of Si12Cr is left empty due to the splitting of 2D shell in D6h symmetry.
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- Gradual variation in the stability of Si12M is related to their electronic structures.
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Electron Counting Rules for Transition Metal-Doped Si12 Clusters Nguyen Duy Phi,a Nguyen Tien Trung,a Ewald Janssens,b Vu Thi Ngana,*
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a) Chemistry Department, Laboratory of Computational Chemistry and Modelling, Quy Nhon University, 170 An Duong Vuong Street, Quy Nhon city, Vietnam
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Email: [email protected]
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b) Laboratory of Solid State Physics and Magnetism, KU Leuven, B-3001 Leuven, Belgium
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Abstract: Application of the phenomenological shell model (PSM) provides an explanation for the enhanced stability of Si12Cr and Si12Fe clusters and relative cluster stability along the Si12M
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(M=Sc-Ni) series. Sequence of orbital shells in PSM is mostly determined by the confining
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potential, which depends on the cluster shape. In D6h hexagonal prism geometry, degenerate 2P and 2D shells undergo splitting, and the energy levels of 2Pz and 2 Dz 2 orbitals become higher 2
y2
), respectively. Therefore, stability of the
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than those of (2Px,2Py) and (2Dxy,2Dyz,2Dxz,2 D x
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most stable Si12Cr and Si12Fe clusters is attributed to the filling of the 2S22P62D8 and 2S22P62D10
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closed shells.
Keywords: phenomenological shell model, doped silicon clusters, electron counting rule, 16-
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electron system, 18-electron system.
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Owing to their potential applications, silicon clusters have been intensively studied over the past three decades, both experimentally and theoretically.[1,2,3,4] The geometry and electronic structure of the Si clusters can be modified by introducing a dopant atom.[5,6,7,8,9] Clusters showing
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enhanced stability compared to neighbouring sizes are regularly referred to as “magic”. The first observation of the magic behaviour in transition metal-doped Si clusters dates back to 1987,
cr
when Beck used laser photoionization and time-of-flight mass spectrometry to discover the
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enhanced stability of Si15M (M=W, Mo, Cr).[5] To the best of our knowledge, almost a decade then passed until the appearance of the first theoretical study on endohedrally doped Si clusters,
an
describing the icosahedral structure of Si20Zr.[6] Since then, endohedrally doped Si clusters have
M
been intensely studied both experimentally and theoretically. Researchers have concentrated on three main issues: (1) the search for new stable doped Si clusters, (2) the determination of the
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magic behaviour of the clusters.
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geometrical and electronic ground state structures of the clusters, and (3) the interpretation of the
For each transition metal dopant, the minimal cluster size for the formation of metal-
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encapsulated silicon cages is called the critical size. This value varies along the 3d series.[10] The smallest size for which all 3d transition metal dopants are expected to assume an endohedral position in the host silicon cluster is Si12. Therefore, the geometric and electronic structures and the stabilities of Si12 doped with different transition metals in various charge states have been extensively studied.[9,11,12,13,14,15,16,17,18] Although experiments did not observe magic behaviour for Si12M clusters, theoretical studies showed that based on energetic criteria, Si12Cr and Si12Fe are more stable than the other clusters in the Si12M series.[18] Some authors have attributed the enhanced stability of Si12Cr in the Si12M (M=ScNi) series to the 18-electron rule,[11,12] while others have cast doubts on the applicability of this electron counting rule.[18]
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In most studies of doped Si clusters, enhanced stabilities are explained by compact structures,[19,20] electronic shell closures,[6] or a combination of both.[11,13] The nearly free electron gas, first applied by Khanna et al. for silicon clusters,[21,22] is the most applied model for
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counting the number of delocalized electrons. However, this model is empirical, as it assumes that all Si atoms binding with the metal dopant atom contribute one valence electron each to the
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free electron gas without considering the details of the electron distribution and the chemical
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bonding in the clusters. As a result, this model is not generally applicable. According to this model, the Si12Cr and Si12Fe clusters are supposed to be magic because they have closed
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electronic shells with 18 and 20 electrons that fill up the 1s2 1p6 1d10 and 1s 21p6 1d10 2s2 orbitals, respectively.
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Very recently, Abreu et al.[15] questioned the applicability of the 18-electron rule for
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Si12Cr, and their analysis of the electronic state of the central Cr atom found that 16 effective
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valence electrons can be assigned to the Cr atom. This model cannot be used to explain the relative stability of the SinFe (n=9-16) because the Fe dopant atom has a fully occupied 3d-shell
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in all of the clusters.[16] In principle, electron counting rules should not rely on the valence electrons of the dopant atom alone because these electrons are only a part of the electron cloud of the cluster. The model of a nearly free electron gas, where each Si atom contributes a single electron to the electron gas of the cluster, is not straightforward either. Instead, the number of electrons taken into account in the counting rules should arise from the number of delocalized electrons over the entire cluster. The aim of the present study is to demonstrate a consistent way to count the number of electrons in the delocalized electron gas in order to explain the relative cluster stability along the Si12M (M=Sc–Ni) series. We first revisit the geometrical properties, electronic structures and
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stability of the Si12M series. Second, the phenomenological shell model (PSM)[23] and the electron localization index (ELI) are applied to count the number of delocalized electrons that fill a selection of the electronic shells.
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The geometric and electronic structures of the clusters are optimized by density functional theory (DFT) using the B3P86 functional in combination with the 6-311+G(d) basis
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set. Frequency calculations are performed at the same level of theory to characterize the
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stationary points on the potential energy surface. All quantum chemical calculations are performed using the Gaussian 03 package.[24] Atomic charges and electron distributions are
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evaluated based on the natural population analysis performed at the B3P86/6-311+G(d) level of theory using the NBO 5.G program.[25]
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Earlier theoretical studies concluded that all Si12M (M=Sc–Ni) clusters adopt hexagonal
d
prism (HP) ground state structures with different degrees of distortion at the lowest spin
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state.[14,21] Experimental evidence for these HP structures is limited. For example, using infrared multiphoton dissociation spectroscopy, Si12V+ and Si12Mn+ cationic clusters were proven to
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exhibit distorted HP structures,[9,17] and photoelectron spectroscopy of the Si12Cr- anion indicated that both the neutral and anionic Si12Cr0,- exhibit HP structures.[26] In both cases, the structural information was obtained by comparison of computational and experimental data. Unfortunately, there is no conclusive evidence about the exact symmetry of the ground state structures of these clusters. Therefore, the HP structures in this work are constrained to different subgroups of the D6h point group to search for the highest possible symmetry of the ground state structures shown in Figure 1. To evaluate their distortions from the ideal D6h symmetric HP, the ground state structures are subsequently constrained to higher symmetries, their geometries re-optimized, and
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their frequencies re-calculated. Characteristics of the ground state structures and their highersymmetry geometries are listed in Table 1. The average Si-Sc bond length in the obtained ground state structure (C1 point group) of
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Si12Sc is 2.88 Å, and the longest bond is 3.14 Å, which is much longer than the bond in the SiSc dimer (2.49 Å). This indicates that the ground state structure of Si12Sc is a very loose cage. In
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addition, the Sc atom seems not to be completely encapsulated by the Si12 cage (cf. Figure 1).
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For these reasons, a previous study concluded that Si12Sc is not a hexagonal prism.[27] Constraining the structure to the Cs symmetry results in a more compact structure than the C1
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structure, with a smaller average bond length of 2.75 Å and the Sc atom completely encapsulated in the cage. The Cs structure is actually a shallow transition state, located 0.12 eV above the
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ground state and possessing an imaginary frequency of 78i cm-1. This implies that the HP
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structure of Si12Sc shows some degree of fluxionality.
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Si12Ti adopts a HP structure at the Cs symmetry (1A’). Its average Si-Ti bond length is 2.68 Å, much shorter than that of the Si12Sc cluster (2.88 Å) but still longer than the bond in the
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SiTi dimer (2.42 Å). Constraining this structure to C2h symmetry (1Ag), a transition state is found that is only 0.03 eV higher in energy than the Cs structure and displays an imaginary frequency of 116i cm-1.
The ground state structure of Si12V is an HP with C2h symmetry (2Ag) and is more compact than those of Si12Sc and Si12Ti. The higher-symmetry D2h structure, with geometry that is nearly the same as the C2h structure, is almost degenerate with the C2h structure (only 0.004 eV higher in energy) and is characterized as a transition state with an imaginary frequency of 85i cm-1.
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Among the studied Si12M (M = Sc–Ni) clusters, the highest symmetry HP (D3d) is obtained for the dopants in the middle of the 3d series (M = Cr, Mn, and Fe). These clusters have the most compact structures in the series, as evidenced by the comparison of the average Si-M
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bond length (2.61 Å, 2.61 Å, and 2.59 Å, respectively). The corresponding ideal D6h structures are slightly higher in energy than the D3d ground states and exhibit a small imaginary frequency,
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characterizing them as transition states. For Si12Cr, the difference in energy between the D6h and
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D3d structures amounts to only 0.001 eV, and the D6h transition state has the smallest imaginary
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frequency (51i cm-1), implying that this cluster is just slightly distorted from the D6h structure. The Si12Co cluster adopts the C2h symmetry in its ground state structure. The average Si-
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Co bond length equals that in Si12Cr, but the longest Si-Co bond (2.86 Å) is much longer, implying that Si12Co has a larger degree of the distortion than Si12Cr. The more symmetric D2h
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structure of Si12Co is a second-order saddle point and lies 0.27 eV above the C2h structure. The
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ground state Cs structure of Si12Ni is even more distorted because the maximal Si-Ni distance is
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as large as 3.7 Å. Its C2h structure is characterized as a second-order saddle point and lies 0.23 eV above the Cs ground state structure. In general, the distortion from the ideal D6h hexagonal prism of the Si12M gradually changes as M goes from Sc to Ni, with the least distortion obtained for the Si12Cr cluster. The strongest distortion is found for Si12Sc and Si12Ni. Based on the presence of low energy transition states with similar structures as the ground states, most Si12M clusters are expected to exhibit some degree of fluxionality. This may result in broad bands of their infrared spectra, as was found experimentally in the case for Si16V+.[17]
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The relative stability along the Si12M (M=Sc–Ni) series is investigated through computed energetic parameters such as the average binding energy (BE), the embedding energy (EE), and the HOMO-LUMO gap. The average binding energy quantifies the strength of the chemical
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bonds in the clusters and is calculated as BE = [12E(Si) + E(M) – E(Si12M)]/13, where E(Si), E(M) and E(Si12M) are the total energies of the Si and M atoms and the Si12M cluster,
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respectively. The embedding energy measures the energy released upon formation of the Si12M
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cluster from the M atom and the Si12 cage. These are computed within the Wigner-Witmer spin conservation rule as EE = E(Si12)+E(M)-E(Si12M), where E(Si12) is the single-point energy of the
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Si12 cage after removing the M atom from the ground state structure of Si12M and E(M) is energy of the M atom in the same spin state as in the Si12M cluster. While the average binding energy
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and the embedding energy are related to the geometrical stability of the clusters, the HOMO-
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LUMO gap provides information about the stability of the electronic structure. The variations of
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these energetic parameters along the Si12M series are illustrated in Figure 2. The different quantities show the same general trend: the stability of clusters changes gradually from Sc to Ni
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with a maximum in the middle of the series and lower stability at the beginning and the end of the series. Figures 2a and 2b show global maxima of BE and EE at M = Cr, meaning that the structural stability is the highest for the Si12Cr cluster. In addition, a local maximum of the BE is found at M = Fe. Changes in the cluster stability appear to be related to the magnitude of the structural distortions from the ideal D6h hexagonal prism. The less-distorted clusters exhibit higher stabilities. The HOMO-LUMO gap indicates the stability of electronic structure; in particular, electronic closure species are often related to a large gap. Figure 2c shows that the largest HOMO-LUMO gap values are obtained for Si12Cr and Si12Fe, suggesting that these have closed electronic shells.
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Based on both the geometric and electronic criteria, the Si12Cr and Si12Fe clusters are found to be the most stable structures in the Si12M (M=Sc-Ni) series. These two clusters also have the most compact structures because they have the highest symmetries and the shortest
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average Si-M bond lengths in the series. Si12Mn shows a slightly lower stability than Si12Cr and
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Si12Fe.
We now analyse the chemical bonding and electron distribution in the clusters to search
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for the connection between their relative stabilities and the electronic structures. We first apply
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the phenomenological shell model (PSM) that has been successfully used to interpret the magic behaviour of the metal clusters.[23,28,29] The main assumption of this model is that the delocalized
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electrons move in an average potential that confines them according to the cluster shape. For a spherical confining potential, the normal shell orbital sequence is 1S, 1P, 1D, 2S, 1F, 2P, 1G,
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2D, 3S,… However, the energy ordering and the degeneracy of the orbitals can be changed by
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both the shape and the charge of the clusters. According to the PSM,[23] metal clusters with
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closed electronic shells, i.e., 1S2, 1S21P6, 1S21P61D10, 1S21P61D102S2… configurations with 2, 8, 18, 20… electrons, are expected to be more stable than the other configurations. Owing to the (distorted) hexagonal prismatic shapes of the Si12M clusters, their orbital shells, which are degenerate in the spherical confining potential such as P, D, F, G, … undergo splitting: P Pz (a2u) + (Px, Py) (e1u); D Dz 2 (a1g) + (Dxz, Dyz) (e1g) + ( D x
2
y2
, Dxy) (e2g); … In
the D6h HP potential, orbitals that are oriented along the z axis, such as Pz and Dz 2 , are energetically destabilized. As a result, the P and D shells in an HP potential can be filled by four or eight electrons, respectively. Higher angular momentum shells, such as F and G, undergo splitting into more subshells. 10 Page 10 of 18
For non-metallic clusters such as the silicon clusters studied in this work, the application of the PSM is difficult because not all of the valence electrons of the Si atoms are delocalized. However, analysis of the electron localizability indicator ELI-D,[30] which can be decomposed
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into partial orbital contributions pELI-D,[31] paves the way for application of the PSM to doped Si clusters. The ELI-D and pELI-D isosurfaces are computed using the DGrid-4.2 program.[32]
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Accordingly, the number of delocalized electrons in the caged SinM clusters can be
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determined.[33] To illustrate the electronic shell structures of these clusters, the total and partial densities of states (DOS and pDOS), which are regarded as a spectrum of the molecular orbitals
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(MOs) and the contribution of the atomic orbitals (AOs) to the MOs, are plotted in Figures 3 and 4 for the ground state structures of the Si12Cr and Si12Fe clusters, respectively. The shapes of the
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MOs are analysed and assigned to the shell orbitals (1S, 1P, 1D, 1F, 2S, 2P, 2D, …) within the
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PSM, as shown in the same figures. The shell sequences derived for Si12Cr and Si12Fe from their
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orbital shapes are as follows:
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Si12Cr: 1S21P41Pz21D81F82S21 Dz22 1F42P41G22D82Pz21G6 2 Dz02 Si12Fe: 1S21P41Pz21D81F82S21 Dz22 1F42D42P41G22D42Pz21G62 Dz22 1G0 These shell sequences show that the Pz subshell is much higher in energy than the Px and Py orbitals. The 2Pz is even higher in energy than the 2Dxy, 2Dyz, 2Dxz and 2 D x 2 Dz 2 is much higher than the 2Dxy, 2Dyz, 2Dxz, and 2 D x
2
y2
2
y2
orbitals and the
orbitals. The N = 2 orbitals of Si12Cr
are filled to the 2D8 subshell, leaving the LUMO-2 Dz 2 empty, while in Si12Fe, the entire 2D10 shell is filled due to the two extra valence electrons of the iron atom. Therefore, according to PSM, both clusters have closed electronic shells that lead to their enhanced stability compared with other clusters in the series. 11 Page 11 of 18
Analysing the partial density of states, we find that all shell orbitals with N = 1 (including 1S, 1P, 1D, 1F and 1G) are composed of the AO-s and AO-p of the Si atoms (red and blue dotted lines, respectively). The shell orbitals with N = 2 (including 2S, 2P and 2D), in contrast, are
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composed of the AOs of both Si and dopant (Cr or Fe) atoms. To reveal the bonding character of each shell orbital group, the pELI-Ds for the two shell orbital groups (N = 1 and N = 2) are
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calculated and their isosurfaces are plotted in Figure 5. For both clusters, the pELI-D domains of
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the first shell orbital group (N = 1), which are plotted in blue, are localized along the Si–Si bonds of the Si12 cage. The pELI-D domains of the second shell orbital group (N = 2), shown in green,
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are pointing out of the cage along the M–Si bonds. These green domains are strongly delocalized over the entire cage. Comparing the pELI-D plots of the Si12Cr and Si12Fe clusters, we find the
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green domains of Si12Fe to be slightly less delocalized than those of Si12Cr. This is because the
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3d orbitals of the Fe atom are more localized than those of the Cr atom. The localization of AO-
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3d is also reflected in the pDOS (green solid lines in Figures 3 and 4). In particular, the pDOS of the AO-3d of the Fe atom is lower in energy than that of the Cr atom. This is related to the fact
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that the Si12Cr is more stable than the Si12Fe cluster. Therefore, the degree of the delocalization can be considered to be a factor determining the stability of clusters. Briefly, the electrons in the shell orbitals with N = 1 are localized along the Si–Si bonds, while those in N = 2 shell orbitals are more delocalized and are responsible for the interactions between the dopant and the cage. Therefore, the latter might be used for an electron count owing to their delocalizing character. Accordingly, Si12Cr has 16 delocalized electrons in filled 2S22P62D8 shells, and Si12Fe has 18 delocalized electrons in filled 2S22P62D10 shells. Thus, both clusters have a stabilized electronic structure and a compact geometry.
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It should be noted that the 1F and 1G shells are not completely filled because these highdegeneracy shells undergo strong splitting in the confining HP potential that is far from spherically shaped. However, that does not weaken the shell closure, as the electrons in the shell
originates from delocalized electrons within the N = 2 shells.
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orbitals with N = 1 are localized on the Si-Si bonds. Conversely, the shell closure of the clusters
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Similar analysis for Si12Mn shows that the electronic structure within PSM is as follows:
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1S21P41Pz21D81F82S21 Dz22 1F42D42P41G22D42Pz21G62 D1z 2 1G0. The Si12Mn cluster is slightly
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less stable than the Si12Cr and Si12Fe clusters because its outmost subshell 2 Dz 2 is half-filled. Thus, the gradual variation of cluster stability in the series should be related to the changes in
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their electronic structures. To clarify this point, their electronic configuration within PSM should be assigned. The shapes of shell orbitals in the other Si12M (M = Sc, Ti, V, Co, Ni) clusters are
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more difficult to assign, as their ground state structures are strongly distorted from the perfect
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D6h symmetry. To overcome this difficulty, we constrained their ground state structures to the
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D6h symmetry, optimized them at a low level of theory (HF/6-31G), and inspected their orbital shapes. We can only obtain the D6h structure for the Si12Ti and Si12V clusters and find that their first shell groups (N = 1) are similar to those of the Si12M (M=Cr, Mn, and Fe), while differences appear in the N = 2 shell groups. In particular, the shell sequences for the second group of the Si12Ti and Si12V clusters are:
Si1 2Ti: 2S2 2P4 2Dxy2 2Dxz2 2Dyz2 2 Dx22 y2 2Pz02 Dz02 Si12V: 2S2 2P4 2Dxy2 2Dxz2 2Dyz2 2 D1x2 y2 2Pz2 2 Dz02 Therefore, the PSM allows us to illustrate the gradual changes in electronic structure of the Si12M clusters with changing M in the 3d series.
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In conclusion, the phenomenological shell model together with the electron localization indicator allows the description of chemical bonding and determines the number of delocalized electrons in the Si12M (M=Sc-Ni) clusters. It is noteworthy that the electrons in the shell orbitals
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with the principle quantum number N = 1 (1S, 1P, 1D, 1F, and 1G) are localized and responsible for the Si–Si bonds of the cage, while the electrons in the shell orbitals with N = 2 (2S, 2P, and
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2D) are delocalized and responsible for the dopant–Si cage interaction. Therefore, we propose to
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use the number of electrons in the shell orbitals with N = 2 for the electron counting. The magic Si12Cr cluster can be reconciled due to the electronic closure up to 2D8 leaving the LUMO-
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2 Dz 2 empty, while the other magic cluster in the series, namely, Si12Fe, is filled to 2D10, resulting
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in 16- and 18-electron systems, respectively. The Si12Cr cluster is more stable than the Si12Fe cluster due to its larger degree of electron delocalization. Application of this model to larger
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stable doped clusters will be the subject of future work and is currently being carried out.
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Table 1. Point group (PG), relative energy (E, eV), electronic state (ES), number of imaginary frequencies (Nimag) and their values, average Si-M bond length (Å), and longest Si-M bond (Å)
clusters.
Si12Ti Si12V Si12Cr Si12Mn Si12Fe Si12Co
A A’ 1 A’ 1 Ag 2 Ag 2 Ag 1 A1g 1 A1g 2 A1g 2 A1g 1 A1g 1 A1g 2 Bg 2 Ag 1 A’ 1 Ag 2
0 1 (78i) 0 1 (116i) 0 1 (85i) 0 1 (51i) 0 1 (71i) 0 1 (123i) 0 2 (88i; 157i) 0 2 (35i; 141i)
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Si12Ni
2
Longest Si-M 3.14 3.12 2.84 2.75 2.68 2.68 2.68 2.61 2.70 2.61 2.74 2.59 2.86 2.64 3.70 3.04
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0.00 0.12 0.00 0.03 0.00 0.004 0.00 0.001 0.00 0.006 0.00 0.09 0.00 0.27 0.00 0.23
Average Si-M 2.88 2.75 2.68 2.67 2.63 2.63 2.61 2.61 2.61 2.61 2.59 2.59 2.61 2.60 2.62 2.61
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C1 Cs Cs C2h C2h D2h D3d D6h D3d D6h D3d D6h C2h D2h Cs C2h
Nimag
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Si12Sc
ES
M
E
d
PG
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Cluster
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of the ground state structures and possible higher-symmetry structures of the Si12M (M = Sc–Ni)
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Figure Captions:
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Figure 1. The lowest-energy strucures of the Si12M (M = Sc-Ni) clusters. Figure 2. Dopant dependence of binding energies (a), embedding energies (b), and HOMO-
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LUMO gaps (c) of Si12M (M = Sc–Ni).
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Figure 3. Total and partial densities of states (DOS, pDOS) of the D3d ground state structure of Si12Cr. Orbital energies are calculated at the B3P86/6-311+G(d) level of theory. Molecular
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orbitals are shown and assigned based on comparison with PSM orbitals.
Figure 4. Total and partial density of states (DOS, pDOS) of the D3d ground state structure of
M
Si12Fe. Orbital energies are calculated at the B3P86/6-311+G(d) level of theory. Molecular orbitals are shown and assigned based on comparison with PSM orbitals.
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Figure 5. Top view (left) and side view (right) of the pELI-D for the D3d ground state structure
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of Si12Cr (a and b) and Si12Fe (c and d). Blue colour displays the 1.8-localization domains for the
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group of the shell orbitals with N = 1 (1S, 1P, 1D, 1F and 1G). Green colour displays the 1.0localization domains for the group of the shell orbitals with N = 2 (2S, 2P and 2D).
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Acknowledgements: This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number “104.06-2013.06”.
[9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
[24] [25]
cr
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an
M
[5] [6] [7] [8]
d
[3] [4]
te
[2]
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W. Zheng, J. M. Nilles, D. Radisic and J. Kit H. Bowen, J. Chem. Phys. 2005, 122, 071101. L. Guo, G. Zhao, Y. Gu, X. Liu and Z. Zeng, Phys. Rev. B 2008, 77, 195417. S. Neukermans, E. Janssens, Z. F. Chen, R. E. Silverans, P. V. Schleyer and P. Lievens, Phys. Rev. Lett. 2004, 92, 163401. T. Höltzl, P. Lievens, T. Veszprémi and M. T. Nguyen, J. Phys. Chem. C 2009, 113, 21016-21018. M. Kohout, F. R. Wagner and Y. Grin, Inter. J. Quant. Chem. 2006, 106, 1499-1507. F. R. Wagner, V. Bezugly, M. Kohout and Y. Grin, Chem. Eur. J. 2007, 13, 5724-5741. M. Kohout in DGrid, Max-Planck Institut fur Chemische Physik und Fester Stoffe, Dresden, 2006. V. T. Ngan and M. T. Nguyen, J. Phys. Chem. A 2010, 114, 7609-7615.
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S0009-2614(15)00876-3 http://dx.doi.org/doi:10.1016/j.cplett.2015.11.025 CPLETT 33429
To appear in: Received date: Revised date: Accepted date:
21-7-2015 12-11-2015 16-11-2015
Please cite this article as: N. Duy, N.T. Trung, E. Janssens, V.T. Ngan, Electron Counting Rules for Transition Metal-Doped Si12 Clusters, Chem. Phys. Lett. (2015), http://dx.doi.org/10.1016/j.cplett.2015.11.025 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Highlights: - PSM and pELI-D allow counting of delocalized electrons for a doped Si cluster.
- The enhanced stability of Si12Cr is due to the 16-electron shell closure. - The enhanced stability of Si12Fe is due to the 18-electron shell closure.
ip t
- 2Dz2 orbital of Si12Cr is left empty due to the splitting of 2D shell in D6h symmetry.
Ac ce p
te
d
M
an
us
cr
- Gradual variation in the stability of Si12M is related to their electronic structures.
1 Page 1 of 18
Electron Counting Rules for Transition Metal-Doped Si12 Clusters Nguyen Duy Phi,a Nguyen Tien Trung,a Ewald Janssens,b Vu Thi Ngana,*
ip t
a) Chemistry Department, Laboratory of Computational Chemistry and Modelling, Quy Nhon University, 170 An Duong Vuong Street, Quy Nhon city, Vietnam
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te
d
M
an
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Email: [email protected]
cr
b) Laboratory of Solid State Physics and Magnetism, KU Leuven, B-3001 Leuven, Belgium
2 Page 2 of 18
Abstract: Application of the phenomenological shell model (PSM) provides an explanation for the enhanced stability of Si12Cr and Si12Fe clusters and relative cluster stability along the Si12M
ip t
(M=Sc-Ni) series. Sequence of orbital shells in PSM is mostly determined by the confining
cr
potential, which depends on the cluster shape. In D6h hexagonal prism geometry, degenerate 2P and 2D shells undergo splitting, and the energy levels of 2Pz and 2 Dz 2 orbitals become higher 2
y2
), respectively. Therefore, stability of the
us
than those of (2Px,2Py) and (2Dxy,2Dyz,2Dxz,2 D x
an
most stable Si12Cr and Si12Fe clusters is attributed to the filling of the 2S22P62D8 and 2S22P62D10
M
closed shells.
Keywords: phenomenological shell model, doped silicon clusters, electron counting rule, 16-
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te
d
electron system, 18-electron system.
3 Page 3 of 18
Owing to their potential applications, silicon clusters have been intensively studied over the past three decades, both experimentally and theoretically.[1,2,3,4] The geometry and electronic structure of the Si clusters can be modified by introducing a dopant atom.[5,6,7,8,9] Clusters showing
ip t
enhanced stability compared to neighbouring sizes are regularly referred to as “magic”. The first observation of the magic behaviour in transition metal-doped Si clusters dates back to 1987,
cr
when Beck used laser photoionization and time-of-flight mass spectrometry to discover the
us
enhanced stability of Si15M (M=W, Mo, Cr).[5] To the best of our knowledge, almost a decade then passed until the appearance of the first theoretical study on endohedrally doped Si clusters,
an
describing the icosahedral structure of Si20Zr.[6] Since then, endohedrally doped Si clusters have
M
been intensely studied both experimentally and theoretically. Researchers have concentrated on three main issues: (1) the search for new stable doped Si clusters, (2) the determination of the
te
magic behaviour of the clusters.
d
geometrical and electronic ground state structures of the clusters, and (3) the interpretation of the
For each transition metal dopant, the minimal cluster size for the formation of metal-
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encapsulated silicon cages is called the critical size. This value varies along the 3d series.[10] The smallest size for which all 3d transition metal dopants are expected to assume an endohedral position in the host silicon cluster is Si12. Therefore, the geometric and electronic structures and the stabilities of Si12 doped with different transition metals in various charge states have been extensively studied.[9,11,12,13,14,15,16,17,18] Although experiments did not observe magic behaviour for Si12M clusters, theoretical studies showed that based on energetic criteria, Si12Cr and Si12Fe are more stable than the other clusters in the Si12M series.[18] Some authors have attributed the enhanced stability of Si12Cr in the Si12M (M=ScNi) series to the 18-electron rule,[11,12] while others have cast doubts on the applicability of this electron counting rule.[18]
4 Page 4 of 18
In most studies of doped Si clusters, enhanced stabilities are explained by compact structures,[19,20] electronic shell closures,[6] or a combination of both.[11,13] The nearly free electron gas, first applied by Khanna et al. for silicon clusters,[21,22] is the most applied model for
ip t
counting the number of delocalized electrons. However, this model is empirical, as it assumes that all Si atoms binding with the metal dopant atom contribute one valence electron each to the
cr
free electron gas without considering the details of the electron distribution and the chemical
us
bonding in the clusters. As a result, this model is not generally applicable. According to this model, the Si12Cr and Si12Fe clusters are supposed to be magic because they have closed
an
electronic shells with 18 and 20 electrons that fill up the 1s2 1p6 1d10 and 1s 21p6 1d10 2s2 orbitals, respectively.
M
Very recently, Abreu et al.[15] questioned the applicability of the 18-electron rule for
d
Si12Cr, and their analysis of the electronic state of the central Cr atom found that 16 effective
te
valence electrons can be assigned to the Cr atom. This model cannot be used to explain the relative stability of the SinFe (n=9-16) because the Fe dopant atom has a fully occupied 3d-shell
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in all of the clusters.[16] In principle, electron counting rules should not rely on the valence electrons of the dopant atom alone because these electrons are only a part of the electron cloud of the cluster. The model of a nearly free electron gas, where each Si atom contributes a single electron to the electron gas of the cluster, is not straightforward either. Instead, the number of electrons taken into account in the counting rules should arise from the number of delocalized electrons over the entire cluster. The aim of the present study is to demonstrate a consistent way to count the number of electrons in the delocalized electron gas in order to explain the relative cluster stability along the Si12M (M=Sc–Ni) series. We first revisit the geometrical properties, electronic structures and
5 Page 5 of 18
stability of the Si12M series. Second, the phenomenological shell model (PSM)[23] and the electron localization index (ELI) are applied to count the number of delocalized electrons that fill a selection of the electronic shells.
ip t
The geometric and electronic structures of the clusters are optimized by density functional theory (DFT) using the B3P86 functional in combination with the 6-311+G(d) basis
cr
set. Frequency calculations are performed at the same level of theory to characterize the
us
stationary points on the potential energy surface. All quantum chemical calculations are performed using the Gaussian 03 package.[24] Atomic charges and electron distributions are
an
evaluated based on the natural population analysis performed at the B3P86/6-311+G(d) level of theory using the NBO 5.G program.[25]
M
Earlier theoretical studies concluded that all Si12M (M=Sc–Ni) clusters adopt hexagonal
d
prism (HP) ground state structures with different degrees of distortion at the lowest spin
te
state.[14,21] Experimental evidence for these HP structures is limited. For example, using infrared multiphoton dissociation spectroscopy, Si12V+ and Si12Mn+ cationic clusters were proven to
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exhibit distorted HP structures,[9,17] and photoelectron spectroscopy of the Si12Cr- anion indicated that both the neutral and anionic Si12Cr0,- exhibit HP structures.[26] In both cases, the structural information was obtained by comparison of computational and experimental data. Unfortunately, there is no conclusive evidence about the exact symmetry of the ground state structures of these clusters. Therefore, the HP structures in this work are constrained to different subgroups of the D6h point group to search for the highest possible symmetry of the ground state structures shown in Figure 1. To evaluate their distortions from the ideal D6h symmetric HP, the ground state structures are subsequently constrained to higher symmetries, their geometries re-optimized, and
6 Page 6 of 18
their frequencies re-calculated. Characteristics of the ground state structures and their highersymmetry geometries are listed in Table 1. The average Si-Sc bond length in the obtained ground state structure (C1 point group) of
ip t
Si12Sc is 2.88 Å, and the longest bond is 3.14 Å, which is much longer than the bond in the SiSc dimer (2.49 Å). This indicates that the ground state structure of Si12Sc is a very loose cage. In
cr
addition, the Sc atom seems not to be completely encapsulated by the Si12 cage (cf. Figure 1).
us
For these reasons, a previous study concluded that Si12Sc is not a hexagonal prism.[27] Constraining the structure to the Cs symmetry results in a more compact structure than the C1
an
structure, with a smaller average bond length of 2.75 Å and the Sc atom completely encapsulated in the cage. The Cs structure is actually a shallow transition state, located 0.12 eV above the
M
ground state and possessing an imaginary frequency of 78i cm-1. This implies that the HP
d
structure of Si12Sc shows some degree of fluxionality.
te
Si12Ti adopts a HP structure at the Cs symmetry (1A’). Its average Si-Ti bond length is 2.68 Å, much shorter than that of the Si12Sc cluster (2.88 Å) but still longer than the bond in the
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SiTi dimer (2.42 Å). Constraining this structure to C2h symmetry (1Ag), a transition state is found that is only 0.03 eV higher in energy than the Cs structure and displays an imaginary frequency of 116i cm-1.
The ground state structure of Si12V is an HP with C2h symmetry (2Ag) and is more compact than those of Si12Sc and Si12Ti. The higher-symmetry D2h structure, with geometry that is nearly the same as the C2h structure, is almost degenerate with the C2h structure (only 0.004 eV higher in energy) and is characterized as a transition state with an imaginary frequency of 85i cm-1.
7 Page 7 of 18
Among the studied Si12M (M = Sc–Ni) clusters, the highest symmetry HP (D3d) is obtained for the dopants in the middle of the 3d series (M = Cr, Mn, and Fe). These clusters have the most compact structures in the series, as evidenced by the comparison of the average Si-M
ip t
bond length (2.61 Å, 2.61 Å, and 2.59 Å, respectively). The corresponding ideal D6h structures are slightly higher in energy than the D3d ground states and exhibit a small imaginary frequency,
cr
characterizing them as transition states. For Si12Cr, the difference in energy between the D6h and
us
D3d structures amounts to only 0.001 eV, and the D6h transition state has the smallest imaginary
an
frequency (51i cm-1), implying that this cluster is just slightly distorted from the D6h structure. The Si12Co cluster adopts the C2h symmetry in its ground state structure. The average Si-
M
Co bond length equals that in Si12Cr, but the longest Si-Co bond (2.86 Å) is much longer, implying that Si12Co has a larger degree of the distortion than Si12Cr. The more symmetric D2h
d
structure of Si12Co is a second-order saddle point and lies 0.27 eV above the C2h structure. The
te
ground state Cs structure of Si12Ni is even more distorted because the maximal Si-Ni distance is
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as large as 3.7 Å. Its C2h structure is characterized as a second-order saddle point and lies 0.23 eV above the Cs ground state structure. In general, the distortion from the ideal D6h hexagonal prism of the Si12M gradually changes as M goes from Sc to Ni, with the least distortion obtained for the Si12Cr cluster. The strongest distortion is found for Si12Sc and Si12Ni. Based on the presence of low energy transition states with similar structures as the ground states, most Si12M clusters are expected to exhibit some degree of fluxionality. This may result in broad bands of their infrared spectra, as was found experimentally in the case for Si16V+.[17]
8 Page 8 of 18
The relative stability along the Si12M (M=Sc–Ni) series is investigated through computed energetic parameters such as the average binding energy (BE), the embedding energy (EE), and the HOMO-LUMO gap. The average binding energy quantifies the strength of the chemical
ip t
bonds in the clusters and is calculated as BE = [12E(Si) + E(M) – E(Si12M)]/13, where E(Si), E(M) and E(Si12M) are the total energies of the Si and M atoms and the Si12M cluster,
cr
respectively. The embedding energy measures the energy released upon formation of the Si12M
us
cluster from the M atom and the Si12 cage. These are computed within the Wigner-Witmer spin conservation rule as EE = E(Si12)+E(M)-E(Si12M), where E(Si12) is the single-point energy of the
an
Si12 cage after removing the M atom from the ground state structure of Si12M and E(M) is energy of the M atom in the same spin state as in the Si12M cluster. While the average binding energy
M
and the embedding energy are related to the geometrical stability of the clusters, the HOMO-
d
LUMO gap provides information about the stability of the electronic structure. The variations of
te
these energetic parameters along the Si12M series are illustrated in Figure 2. The different quantities show the same general trend: the stability of clusters changes gradually from Sc to Ni
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with a maximum in the middle of the series and lower stability at the beginning and the end of the series. Figures 2a and 2b show global maxima of BE and EE at M = Cr, meaning that the structural stability is the highest for the Si12Cr cluster. In addition, a local maximum of the BE is found at M = Fe. Changes in the cluster stability appear to be related to the magnitude of the structural distortions from the ideal D6h hexagonal prism. The less-distorted clusters exhibit higher stabilities. The HOMO-LUMO gap indicates the stability of electronic structure; in particular, electronic closure species are often related to a large gap. Figure 2c shows that the largest HOMO-LUMO gap values are obtained for Si12Cr and Si12Fe, suggesting that these have closed electronic shells.
9 Page 9 of 18
Based on both the geometric and electronic criteria, the Si12Cr and Si12Fe clusters are found to be the most stable structures in the Si12M (M=Sc-Ni) series. These two clusters also have the most compact structures because they have the highest symmetries and the shortest
ip t
average Si-M bond lengths in the series. Si12Mn shows a slightly lower stability than Si12Cr and
cr
Si12Fe.
We now analyse the chemical bonding and electron distribution in the clusters to search
us
for the connection between their relative stabilities and the electronic structures. We first apply
an
the phenomenological shell model (PSM) that has been successfully used to interpret the magic behaviour of the metal clusters.[23,28,29] The main assumption of this model is that the delocalized
M
electrons move in an average potential that confines them according to the cluster shape. For a spherical confining potential, the normal shell orbital sequence is 1S, 1P, 1D, 2S, 1F, 2P, 1G,
d
2D, 3S,… However, the energy ordering and the degeneracy of the orbitals can be changed by
te
both the shape and the charge of the clusters. According to the PSM,[23] metal clusters with
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closed electronic shells, i.e., 1S2, 1S21P6, 1S21P61D10, 1S21P61D102S2… configurations with 2, 8, 18, 20… electrons, are expected to be more stable than the other configurations. Owing to the (distorted) hexagonal prismatic shapes of the Si12M clusters, their orbital shells, which are degenerate in the spherical confining potential such as P, D, F, G, … undergo splitting: P Pz (a2u) + (Px, Py) (e1u); D Dz 2 (a1g) + (Dxz, Dyz) (e1g) + ( D x
2
y2
, Dxy) (e2g); … In
the D6h HP potential, orbitals that are oriented along the z axis, such as Pz and Dz 2 , are energetically destabilized. As a result, the P and D shells in an HP potential can be filled by four or eight electrons, respectively. Higher angular momentum shells, such as F and G, undergo splitting into more subshells. 10 Page 10 of 18
For non-metallic clusters such as the silicon clusters studied in this work, the application of the PSM is difficult because not all of the valence electrons of the Si atoms are delocalized. However, analysis of the electron localizability indicator ELI-D,[30] which can be decomposed
ip t
into partial orbital contributions pELI-D,[31] paves the way for application of the PSM to doped Si clusters. The ELI-D and pELI-D isosurfaces are computed using the DGrid-4.2 program.[32]
cr
Accordingly, the number of delocalized electrons in the caged SinM clusters can be
us
determined.[33] To illustrate the electronic shell structures of these clusters, the total and partial densities of states (DOS and pDOS), which are regarded as a spectrum of the molecular orbitals
an
(MOs) and the contribution of the atomic orbitals (AOs) to the MOs, are plotted in Figures 3 and 4 for the ground state structures of the Si12Cr and Si12Fe clusters, respectively. The shapes of the
M
MOs are analysed and assigned to the shell orbitals (1S, 1P, 1D, 1F, 2S, 2P, 2D, …) within the
d
PSM, as shown in the same figures. The shell sequences derived for Si12Cr and Si12Fe from their
te
orbital shapes are as follows:
Ac ce p
Si12Cr: 1S21P41Pz21D81F82S21 Dz22 1F42P41G22D82Pz21G6 2 Dz02 Si12Fe: 1S21P41Pz21D81F82S21 Dz22 1F42D42P41G22D42Pz21G62 Dz22 1G0 These shell sequences show that the Pz subshell is much higher in energy than the Px and Py orbitals. The 2Pz is even higher in energy than the 2Dxy, 2Dyz, 2Dxz and 2 D x 2 Dz 2 is much higher than the 2Dxy, 2Dyz, 2Dxz, and 2 D x
2
y2
2
y2
orbitals and the
orbitals. The N = 2 orbitals of Si12Cr
are filled to the 2D8 subshell, leaving the LUMO-2 Dz 2 empty, while in Si12Fe, the entire 2D10 shell is filled due to the two extra valence electrons of the iron atom. Therefore, according to PSM, both clusters have closed electronic shells that lead to their enhanced stability compared with other clusters in the series. 11 Page 11 of 18
Analysing the partial density of states, we find that all shell orbitals with N = 1 (including 1S, 1P, 1D, 1F and 1G) are composed of the AO-s and AO-p of the Si atoms (red and blue dotted lines, respectively). The shell orbitals with N = 2 (including 2S, 2P and 2D), in contrast, are
ip t
composed of the AOs of both Si and dopant (Cr or Fe) atoms. To reveal the bonding character of each shell orbital group, the pELI-Ds for the two shell orbital groups (N = 1 and N = 2) are
cr
calculated and their isosurfaces are plotted in Figure 5. For both clusters, the pELI-D domains of
us
the first shell orbital group (N = 1), which are plotted in blue, are localized along the Si–Si bonds of the Si12 cage. The pELI-D domains of the second shell orbital group (N = 2), shown in green,
an
are pointing out of the cage along the M–Si bonds. These green domains are strongly delocalized over the entire cage. Comparing the pELI-D plots of the Si12Cr and Si12Fe clusters, we find the
M
green domains of Si12Fe to be slightly less delocalized than those of Si12Cr. This is because the
d
3d orbitals of the Fe atom are more localized than those of the Cr atom. The localization of AO-
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3d is also reflected in the pDOS (green solid lines in Figures 3 and 4). In particular, the pDOS of the AO-3d of the Fe atom is lower in energy than that of the Cr atom. This is related to the fact
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that the Si12Cr is more stable than the Si12Fe cluster. Therefore, the degree of the delocalization can be considered to be a factor determining the stability of clusters. Briefly, the electrons in the shell orbitals with N = 1 are localized along the Si–Si bonds, while those in N = 2 shell orbitals are more delocalized and are responsible for the interactions between the dopant and the cage. Therefore, the latter might be used for an electron count owing to their delocalizing character. Accordingly, Si12Cr has 16 delocalized electrons in filled 2S22P62D8 shells, and Si12Fe has 18 delocalized electrons in filled 2S22P62D10 shells. Thus, both clusters have a stabilized electronic structure and a compact geometry.
12 Page 12 of 18
It should be noted that the 1F and 1G shells are not completely filled because these highdegeneracy shells undergo strong splitting in the confining HP potential that is far from spherically shaped. However, that does not weaken the shell closure, as the electrons in the shell
originates from delocalized electrons within the N = 2 shells.
ip t
orbitals with N = 1 are localized on the Si-Si bonds. Conversely, the shell closure of the clusters
cr
Similar analysis for Si12Mn shows that the electronic structure within PSM is as follows:
us
1S21P41Pz21D81F82S21 Dz22 1F42D42P41G22D42Pz21G62 D1z 2 1G0. The Si12Mn cluster is slightly
an
less stable than the Si12Cr and Si12Fe clusters because its outmost subshell 2 Dz 2 is half-filled. Thus, the gradual variation of cluster stability in the series should be related to the changes in
M
their electronic structures. To clarify this point, their electronic configuration within PSM should be assigned. The shapes of shell orbitals in the other Si12M (M = Sc, Ti, V, Co, Ni) clusters are
d
more difficult to assign, as their ground state structures are strongly distorted from the perfect
te
D6h symmetry. To overcome this difficulty, we constrained their ground state structures to the
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D6h symmetry, optimized them at a low level of theory (HF/6-31G), and inspected their orbital shapes. We can only obtain the D6h structure for the Si12Ti and Si12V clusters and find that their first shell groups (N = 1) are similar to those of the Si12M (M=Cr, Mn, and Fe), while differences appear in the N = 2 shell groups. In particular, the shell sequences for the second group of the Si12Ti and Si12V clusters are:
Si1 2Ti: 2S2 2P4 2Dxy2 2Dxz2 2Dyz2 2 Dx22 y2 2Pz02 Dz02 Si12V: 2S2 2P4 2Dxy2 2Dxz2 2Dyz2 2 D1x2 y2 2Pz2 2 Dz02 Therefore, the PSM allows us to illustrate the gradual changes in electronic structure of the Si12M clusters with changing M in the 3d series.
13 Page 13 of 18
In conclusion, the phenomenological shell model together with the electron localization indicator allows the description of chemical bonding and determines the number of delocalized electrons in the Si12M (M=Sc-Ni) clusters. It is noteworthy that the electrons in the shell orbitals
ip t
with the principle quantum number N = 1 (1S, 1P, 1D, 1F, and 1G) are localized and responsible for the Si–Si bonds of the cage, while the electrons in the shell orbitals with N = 2 (2S, 2P, and
cr
2D) are delocalized and responsible for the dopant–Si cage interaction. Therefore, we propose to
us
use the number of electrons in the shell orbitals with N = 2 for the electron counting. The magic Si12Cr cluster can be reconciled due to the electronic closure up to 2D8 leaving the LUMO-
an
2 Dz 2 empty, while the other magic cluster in the series, namely, Si12Fe, is filled to 2D10, resulting
M
in 16- and 18-electron systems, respectively. The Si12Cr cluster is more stable than the Si12Fe cluster due to its larger degree of electron delocalization. Application of this model to larger
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te
d
stable doped clusters will be the subject of future work and is currently being carried out.
14 Page 14 of 18
Table 1. Point group (PG), relative energy (E, eV), electronic state (ES), number of imaginary frequencies (Nimag) and their values, average Si-M bond length (Å), and longest Si-M bond (Å)
clusters.
Si12Ti Si12V Si12Cr Si12Mn Si12Fe Si12Co
A A’ 1 A’ 1 Ag 2 Ag 2 Ag 1 A1g 1 A1g 2 A1g 2 A1g 1 A1g 1 A1g 2 Bg 2 Ag 1 A’ 1 Ag 2
0 1 (78i) 0 1 (116i) 0 1 (85i) 0 1 (51i) 0 1 (71i) 0 1 (123i) 0 2 (88i; 157i) 0 2 (35i; 141i)
Ac ce p
Si12Ni
2
Longest Si-M 3.14 3.12 2.84 2.75 2.68 2.68 2.68 2.61 2.70 2.61 2.74 2.59 2.86 2.64 3.70 3.04
cr
0.00 0.12 0.00 0.03 0.00 0.004 0.00 0.001 0.00 0.006 0.00 0.09 0.00 0.27 0.00 0.23
Average Si-M 2.88 2.75 2.68 2.67 2.63 2.63 2.61 2.61 2.61 2.61 2.59 2.59 2.61 2.60 2.62 2.61
us
C1 Cs Cs C2h C2h D2h D3d D6h D3d D6h D3d D6h C2h D2h Cs C2h
Nimag
an
Si12Sc
ES
M
E
d
PG
te
Cluster
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of the ground state structures and possible higher-symmetry structures of the Si12M (M = Sc–Ni)
15 Page 15 of 18
Figure Captions:
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Figure 1. The lowest-energy strucures of the Si12M (M = Sc-Ni) clusters. Figure 2. Dopant dependence of binding energies (a), embedding energies (b), and HOMO-
cr
LUMO gaps (c) of Si12M (M = Sc–Ni).
us
Figure 3. Total and partial densities of states (DOS, pDOS) of the D3d ground state structure of Si12Cr. Orbital energies are calculated at the B3P86/6-311+G(d) level of theory. Molecular
an
orbitals are shown and assigned based on comparison with PSM orbitals.
Figure 4. Total and partial density of states (DOS, pDOS) of the D3d ground state structure of
M
Si12Fe. Orbital energies are calculated at the B3P86/6-311+G(d) level of theory. Molecular orbitals are shown and assigned based on comparison with PSM orbitals.
d
Figure 5. Top view (left) and side view (right) of the pELI-D for the D3d ground state structure
te
of Si12Cr (a and b) and Si12Fe (c and d). Blue colour displays the 1.8-localization domains for the
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group of the shell orbitals with N = 1 (1S, 1P, 1D, 1F and 1G). Green colour displays the 1.0localization domains for the group of the shell orbitals with N = 2 (2S, 2P and 2D).
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Acknowledgements: This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number “104.06-2013.06”.
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