Electron counting rules for transition metal-doped ...

Chemical Physics Letters 643 (2016) 103–108

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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 Ngan a,∗ a b

Chemistry Department, Laboratory of Computational Chemistry and Modelling, Quy Nhon University, 170 An Duong Vuong Street, Quy Nhon City, Vietnam Laboratory of Solid State Physics and Magnetism, KU Leuven, B-3001 Leuven, Belgium

a r t i c l e

i n f o

Article history: Received 21 July 2015 In final form 16 November 2015 Available online 23 November 2015

a b s t r a c t Application of the phenomenological shell model (PSM) provides an explanation for the enhanced stability of Si12 Cr and Si12 Fe clusters and relative cluster stability along the Si12 M (M = Sc Ni) series. Sequence of orbital shells in PSM is mostly determined by the confining 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 2Dz2 orbitals become higher than those of (2Px , 2Py ) and (2Dxy , 2Dyz , 2Dxz , 2Dx2 −y2 ), respectively. Therefore, stability of the most stable Si12 Cr and Si12 Fe clusters is attributed to the filling of the 2S2 2P6 2D8 and 2S2 2P6 2D10 closed shells. © 2015 Elsevier B.V. All rights reserved.

Owing to their potential applications, silicon clusters have been intensively studied over the past three decades, both experimentally and theoretically [1–4]. The geometry and electronic structure of the Si clusters can be modified by introducing a dopant atom [5–9]. Clusters showing 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, when Beck used laser photoionization and time-of-flight mass spectrometry to discover the enhanced stability of Si15 M (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, describing the icosahedral structure of Si20 Zr [6]. Since then, endohedrally doped Si clusters have 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 geometrical and electronic ground state structures of the clusters; and (3) the interpretation of the magic behaviour of the clusters. For each transition metal dopant, the minimal cluster size for the formation of metal-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

∗ Corresponding author. E-mail address: [email protected] (V.T. Ngan). http://dx.doi.org/10.1016/j.cplett.2015.11.025 0009-2614/© 2015 Elsevier B.V. All rights reserved.

have been extensively studied [9,11–18]. Although experiments did not observe magic behaviour for Si12 M clusters, theoretical studies showed that based on energetic criteria, Si12 Cr and Si12 Fe are more stable than the other clusters in the Si12 M series [18]. Some authors have attributed the enhanced stability of Si12 Cr in the Si12 M (M = Sc Ni) series to the 18-electron rule [11,12], while others have cast doubts on the applicability of this electron counting rule [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 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 free electron gas without considering the details of the electron distribution and the chemical bonding in the clusters. As a result, this model is not generally applicable. According to this model, the Si12 Cr and Si12 Fe clusters are supposed to be magic because they have closed electronic shells with 18 and 20 electrons that fill up the 1s2 1p6 1d10 and 1s 2 1p6 1d10 2s2 orbitals, respectively. Very recently, Abreu et al. [15] questioned the applicability of the 18-electron rule for Si12 Cr, and their analysis of the electronic state of the central Cr atom found that 16 effective valence electrons can be assigned to the Cr atom. This model cannot be used to explain the relative stability of the Sin Fe (n = 9–16) because the Fe dopant atom has a fully occupied 3d-shell 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

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Figure 1. The lowest-energy structures of the Si12 M (M = Sc Ni) clusters.

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 Si12 M (M = Sc Ni) series. We first revisit the geometrical properties, electronic structures and stability of the Si12 M 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. 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 set. Frequency calculations are performed at the same level of theory to characterize the 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 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]. Earlier theoretical studies concluded that all Si12 M (M = Sc Ni) clusters adopt hexagonal prism (HP) ground state structures with different degrees of distortion at the lowest spin state [14,21]. Experimental evidence for these HP structures is limited. For example, using infrared multiphoton dissociation spectroscopy, Si12 V+ and Si12 Mn+ cationic clusters were proven to exhibit distorted HP structures [9,17], and photoelectron spectroscopy of the Si12 Cr− anion indicated that both the neutral and anionic Si12 Cr0,− 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 their frequencies re-calculated. Characteristics of the ground state structures and their higher-symmetry geometries are listed in Table 1. The average Si-Sc bond length in the obtained ground state ˚ and the longest bond structure (C1 point group) of Si12 Sc is 2.88 A, ˚ which is much longer than the bond in the SiSc dimer is 3.14 A, ˚ This indicates that the ground state structure of Si12 Sc is a (2.49 A). very loose cage. In addition, the Sc atom seems not to be completely encapsulated by the Si12 cage (cf. Figure 1). For these reasons, a previous study concluded that Si12 Sc is not a hexagonal prism [27]. Constraining the structure to the Cs symmetry results in a more

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) of the ground state structures and possible higher-symmetry structures of the Si12 M (M = Sc Ni) clusters. Cluster

PG

E

ES

Nimag

Average Si M

Longest Si M

Si12 Sc

C1 Cs Cs C2h C2h D2h D3d D6h D3d D6h D3d D6h C2h D2h Cs C2h

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

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)

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

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

Si12 Ti Si12 V Si12 Cr Si12 Mn Si12 Fe Si12 Co Si12 Ni

A 2 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

compact structure than the C1 structure, with a smaller average bond length of 2.75 A˚ and the Sc atom completely encapsulated in the cage. The Cs structure is actually a shallow transition state, located 0.12 eV above the ground state and possessing an imaginary frequency of 78i cm−1 . This implies that the HP structure of Si12 Sc shows some degree of fluxionality. Si12 Ti adopts a HP structure at the Cs symmetry (1 A ). Its average ˚ much shorter than that of the Si12 Sc Si Ti bond length is 2.68 A, ˚ but still longer than the bond in the SiTi dimer cluster (2.88 A) ˚ Constraining this structure to C2h symmetry (1 Ag ), a tran(2.42 A). sition 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 Si12 V is an HP with C2h symmetry (2 Ag ) and is more compact than those of Si12 Sc and Si12 Ti. 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 . Among the studied Si12 M (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 ˚ 2.61 A, ˚ and 2.59 A, ˚ respecof the average Si M bond length (2.61 A, tively). The corresponding ideal D6h structures are slightly higher in energy than the D3d ground states and exhibit a small imaginary frequency, characterizing them as transition states. For Si12 Cr, the difference in energy between the D6h and D3d structures amounts to only 0.001 eV, and the D6h transition state has the smallest


imaginary frequency (51i cm−1 ), implying that this cluster is just slightly distorted from the D6h structure. The Si12 Co cluster adopts the C2h symmetry in its ground state structure. The average Si Co bond length equals that in Si12 Cr, ˚ is much longer, implying that but the longest Si Co bond (2.86 A) Si12 Co has a larger degree of the distortion than Si12 Cr. The more symmetric D2h structure of Si12 Co is a second-order saddle point and lies 0.27 eV above the C2h structure. The ground state Cs structure of Si12 Ni is even more distorted because the maximal Si Ni ˚ Its C2h structure is characterized as distance is as large as 3.7 A. 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 Si12 M gradually changes as M goes from Sc to Ni, with the least distortion obtained for the Si12 Cr cluster. The strongest distortion is found for Si12 Sc and Si12 Ni. Based on the presence of low energy transition states with similar structures as the ground states, most Si12 M 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 Si16 V+ [17]. The relative stability along the Si12 M (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 bonds in the clusters and is calculated as BE = [12E(Si) + E(M) − E(Si12 M)]/13, where E(Si), E(M) and E(Si12 M) are the total energies of the Si and M atoms and the Si12 M cluster, respectively. The embedding energy measures the energy released upon formation of the Si12 M 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(Si12 M), where E(Si12 ) is the single-point energy of the Si12 cage after removing the M atom from the ground state structure of Si12 M and E(M) is energy of the M atom in the same spin state as in the Si12 M cluster. While the average binding energy and the embedding energy are related to the geometrical stability of the clusters, the HOMO-LUMO gap provides information about the stability of the electronic structure. The variations of these energetic parameters along the Si12 M series are illustrated in Figure 2. The different quantities show the same general trend: the stability of clusters changes gradually from Sc to Ni with a maximum in the middle of the series and lower stability at the beginning and the end of the series. Figure 2a and b shows global maxima of BE and EE at M = Cr, meaning that the structural stability is the highest for the Si12 Cr 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 Si12 Cr and Si12 Fe, suggesting that these have closed electronic shells. Based on both the geometric and electronic criteria, the Si12 Cr and Si12 Fe clusters are found to be the most stable structures in the Si12 M (M = Sc Ni) series. These two clusters also have the most compact structures because they have the highest symmetries and the shortest average Si M bond lengths in the series. Si12 Mn shows a slightly lower stability than Si12 Cr and Si12 Fe. We now analyze the chemical bonding and electron distribution in the clusters to search for the connection between their relative stabilities and the electronic structures. We first apply 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 electrons move in an average potential that confines them according to the cluster shape. For a spherical confining potential, the normal shell

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Figure 2. Dopant dependence of binding energies (a), embedding energies (b), and HOMO-LUMO gaps (c) of Si12 M (M = Sc Ni).

orbital sequence is 1S, 1P, 1D, 2S, 1F, 2P, 1G, 2D, 3S, . . . However, the energy ordering and the degeneracy of the orbitals can be changed by both the shape and the charge of the clusters. According to the PSM [23], metal clusters with closed electronic shells, i.e., 1S2 , 1S2 1P6 , 1S2 1P6 1D10 , 1S2 1P6 1D10 2S2 , . . . 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 Si12 M 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 → Dz2 (a1g ) + (Dxz , Dyz )(e1g ) + (Dx2 −y2 , Dxy )(e2g ); . . . In the D6h HP potential, orbitals that are oriented along the z axis, such as Pz and Dz2 , 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. For non-metallic clusters such as the silicon clusters studied in this work, the application of the PSM is difficult because not all

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The shell sequences derived for Si12 Cr and Si12 Fe from their orbital shapes are as follows: Si12 Cr : 1S 2 1P 4 1Pz2 1D8 1F 8 2S 2 1Dz22 1F 4 2P 4 1G2 2D8 2Pz 2 1G6 2Dz02 Si12 Fe : 1S 2 1P 4 1Pz2 1D8 1F 8 2S 2 1Dz22 1F 4 2D4 2P 4 1G2 2D4 2Pz 2 1G6 2Dz22 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 2Dx2 −y2 orbitals and the 2Dz2 is much higher than the 2Dxy , 2Dyz , 2Dxz , and 2Dx2 −y2 orbitals. The

Figure 3. Total and partial densities of states (DOS, pDOS) of the D3d ground state structure of Si12 Cr. 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 (For interpretation of the references to colour in the text, the reader is referred to the web version of this article.).

Figure 4. Total and partial density of states (DOS, pDOS) of the D3d ground state structure of Si12 Fe. 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 (For interpretation of the references to colour in the text, the reader is referred to the web version of this article.).

of the valence electrons of the Si atoms are delocalized. However, analysis of the electron localizability indicator ELI-D [30], which can be decomposed 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]. Accordingly, the number of delocalized electrons in the caged Sin M clusters can be 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 (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 Si12 Cr and Si12 Fe clusters, respectively. The shapes of the MOs are analyzed and assigned to the shell orbitals (1S, 1P, 1D, 1F, 2S, 2P, 2D, . . .) within the PSM, as shown in the same figures.

N = 2 orbitals of Si12 Cr are filled to the 2D8 subshell, leaving the LUMO-2Dz2 empty, while in Si12 Fe, 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. 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 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 calculated and their isosurfaces are plotted in Figure 5. For both clusters, the pELI-D domains of 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, 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 Si12 Cr and Si12 Fe clusters, we find the green domains of Si12 Fe to be slightly less delocalized than those of Si12 Cr. This is because the 3d orbitals of the Fe atom are more localized than those of the Cr atom. The localization of AO-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 that the Si12 Cr is more stable than the Si12 Fe 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, Si12 Cr has 16 delocalized electrons in filled 2S2 2P6 2D8 shells, and Si12 Fe has 18 delocalized electrons in filled 2S2 2P6 2D10 shells. Thus, both clusters have a stabilized electronic structure and a compact geometry. It should be noted that the 1F and 1G shells are not completely filled because these high-degeneracy 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 orbitals with N = 1 are localized on the Si Si bonds. Conversely, the shell closure of the clusters originates from delocalized electrons within the N = 2 shells. Similar analysis for Si12 Mn shows that the electronic structure within PSM is as follows: Si12 Cr : 1S 2 1P 4 1Pz2 1D8 1F 8 2S 2 1D22 1F 4 2P 4 1G2 2D4 2Pz 2 1G6 2D12 1G0 . The z z Si12 Mn cluster is slightly less stable than the Si12 Cr and Si12 Fe clusters because its outmost subshell 2Dz2 is half-filled. Thus, the gradual variation of cluster stability in the series should be related to the changes in their electronic structures. To clarify


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Figure 5. Top view (left) and side view (right) of the pELI-D for the D3d ground state structure of Si12 Cr (a and b) and Si12 Fe (c and d). Blue colour displays the 1.8-localization domains for the group of the shell orbitals with N = 1 (1S, 1P, 1D, 1F and 1G). Green colour displays the 1.0-localization domains for the group of the shell orbitals with N = 2 (2S, 2P and 2D) (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.).

this point, their electronic configuration within PSM should be assigned. The shapes of shell orbitals in the other Si12 M (M = Sc, Ti, V, Co, Ni) clusters are more difficult to assign, as their ground state structures are strongly distorted from the perfect D6h symmetry. To overcome this difficulty, we constrained their ground state structures to the 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 Si12 Ti and Si12 V clusters and find that their first shell groups (N = 1) are similar to those of the Si12 M (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 Si12 Ti and Si12 V clusters are: Si12 Ti : 2S 2 2P 4 2Dxy 2 2Dxz 2 2Dyz 2 2Dx22 −y2 2Pz 0 Dz02 Si12 V : 2S 2 2P 4 2Dxy 2 2Dxz 2 2Dyz 2 2Dx12 −y2 2P2z Dz02 Therefore, the PSM allows us to illustrate the gradual changes in electronic structure of the Si12 M clusters with changing M in the 3d series. 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 Si12 M (M = Sc Ni) clusters. It is noteworthy that the electrons in the shell orbitals 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 2D) are delocalized and responsible for the dopant–Si cage interaction. Therefore, we propose to use the number of electrons in the shell orbitals with N = 2 for the electron counting. The magic Si12 Cr cluster can be reconciled due to the electronic closure up to 2D8 leaving the LUMO-2Dz2 empty, while the other magic cluster in the series, namely, Si12 Fe, is filled to 2D10 , resulting in 16and 18-electron systems, respectively. The Si12 Cr cluster is more stable than the Si12 Fe cluster due to its larger degree of electron

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